# Circle Boundaries and Zero-Radius Foundations

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261005-advance-20261006-r05-circle-foundations · 2026-10-05 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-12, 5) and radius r=0≥0. For P=(-12, 5), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 0 with r²=0; equality is required, giving Yes. Strictly inside points are not boundary members. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0001/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 2. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 4) and radius r=0≥0. For P=(-10, 4), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 1 with r²=0; equality is required, giving No. Strictly inside points are not boundary members. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0002/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 3. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 3) and radius r=1≥0. For P=(-8, 4), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 5 with r²=1; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0003/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 4. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 2) and radius r=2≥0. For P=(-7, 2), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 4 with r²=4; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0004/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 5. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, 1) and radius r=3≥0. For P=(-8, 3), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 4 with r²=9; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0005/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 6. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, 0) and radius r=4≥0. For P=(-2, 1), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 26 with r²=16; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0006/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 7. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. For P=(-1, -1), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 25 with r²=25; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0007/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 8. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -2) and radius r=6≥0. For P=(-5, 3), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 25 with r²=36; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0008/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 9. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -3) and radius r=7≥0. For P=(4, -2), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 65 with r²=49; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0009/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 10. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -4) and radius r=8≥0. For P=(5, -4), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 64 with r²=64; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0010/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 11. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -5) and radius r=9≥0. For P=(-2, 3), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 64 with r²=81; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0011/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 12. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. For P=(10, -5), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 122 with r²=100; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0012/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 13. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -7) and radius r=11≥0. For P=(11, -7), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 121 with r²=121; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0013/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 14. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -8) and radius r=12≥0. For P=(1, 3), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 121 with r²=144; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0014/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 15. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -9) and radius r=13≥0. For P=(16, -8), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 197 with r²=169; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0015/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 16. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -10) and radius r=14≥0. For P=(17, -10), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 196 with r²=196; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0016/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 17. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -11) and radius r=15≥0. For P=(4, 3), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 196 with r²=225; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0017/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 18. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -12) and radius r=16≥0. For P=(22, -11), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 290 with r²=256; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0018/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 19. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -13) and radius r=17≥0. For P=(23, -13), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 289 with r²=289; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0019/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 20. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -14) and radius r=18≥0. For P=(7, 3), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 289 with r²=324; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0020/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 21. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -15) and radius r=19≥0. For P=(28, -14), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 401 with r²=361; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0021/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 22. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. For P=(29, -16), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 400 with r²=400; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0022/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 23. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -17) and radius r=21≥0. For P=(10, 3), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 400 with r²=441; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0023/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 24. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -18) and radius r=22≥0. For P=(34, -17), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationNo

Compare squared distance 530 with r²=484; equality is required, giving No. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0024/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 25. Membership in a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. For P=(35, -19), is P a member of the circle boundary, not merely of the filled disk?

View answer and explanationYes

Compare squared distance 529 with r²=529; equality is required, giving Yes. Strictly inside points are not boundary members.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0025/)

Circle boundaries and zero-radius degeneracy · qf-circle-membership · candidate





## 26. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-12, 5) and radius r=0≥0. Classify P=(-12, 5) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 0 with r²=0: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0026/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 27. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 4) and radius r=0≥0. Classify P=(-10, 4) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 1 with r²=0: less means inside, equal means on boundary, and greater means outside. Here the result is outside. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0027/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 28. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 3) and radius r=1≥0. Classify P=(-8, 4) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 5 with r²=1: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0028/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 29. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 2) and radius r=2≥0. Classify P=(-7, 2) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 4 with r²=4: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0029/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 30. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, 1) and radius r=3≥0. Classify P=(-8, 3) as inside, on boundary, or outside.

View answer and explanationinside

Compare squared distance 4 with r²=9: less means inside, equal means on boundary, and greater means outside. Here the result is inside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0030/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 31. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, 0) and radius r=4≥0. Classify P=(-2, 1) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 26 with r²=16: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0031/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 32. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. Classify P=(-1, -1) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 25 with r²=25: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0032/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 33. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -2) and radius r=6≥0. Classify P=(-5, 3) as inside, on boundary, or outside.

View answer and explanationinside

Compare squared distance 25 with r²=36: less means inside, equal means on boundary, and greater means outside. Here the result is inside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0033/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 34. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -3) and radius r=7≥0. Classify P=(4, -2) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 65 with r²=49: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0034/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 35. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -4) and radius r=8≥0. Classify P=(5, -4) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 64 with r²=64: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0035/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 36. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -5) and radius r=9≥0. Classify P=(-2, 3) as inside, on boundary, or outside.

View answer and explanationinside

Compare squared distance 64 with r²=81: less means inside, equal means on boundary, and greater means outside. Here the result is inside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0036/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 37. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -6) and radius r=10≥0. Classify P=(10, -5) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 122 with r²=100: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0037/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 38. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -7) and radius r=11≥0. Classify P=(11, -7) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 121 with r²=121: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0038/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 39. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -8) and radius r=12≥0. Classify P=(1, 3) as inside, on boundary, or outside.

View answer and explanationinside

Compare squared distance 121 with r²=144: less means inside, equal means on boundary, and greater means outside. Here the result is inside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0039/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 40. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -9) and radius r=13≥0. Classify P=(16, -8) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 197 with r²=169: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0040/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 41. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -10) and radius r=14≥0. Classify P=(17, -10) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 196 with r²=196: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0041/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 42. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -11) and radius r=15≥0. Classify P=(4, 3) as inside, on boundary, or outside.

View answer and explanationinside

Compare squared distance 196 with r²=225: less means inside, equal means on boundary, and greater means outside. Here the result is inside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0042/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 43. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -12) and radius r=16≥0. Classify P=(22, -11) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 290 with r²=256: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0043/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 44. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -13) and radius r=17≥0. Classify P=(23, -13) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 289 with r²=289: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0044/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 45. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -14) and radius r=18≥0. Classify P=(7, 3) as inside, on boundary, or outside.

View answer and explanationinside

Compare squared distance 289 with r²=324: less means inside, equal means on boundary, and greater means outside. Here the result is inside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0045/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 46. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -15) and radius r=19≥0. Classify P=(28, -14) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 401 with r²=361: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0046/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 47. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -16) and radius r=20≥0. Classify P=(29, -16) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 400 with r²=400: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0047/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 48. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -17) and radius r=21≥0. Classify P=(10, 3) as inside, on boundary, or outside.

View answer and explanationinside

Compare squared distance 400 with r²=441: less means inside, equal means on boundary, and greater means outside. Here the result is inside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0048/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 49. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -18) and radius r=22≥0. Classify P=(34, -17) as inside, on boundary, or outside.

View answer and explanationoutside

Compare squared distance 530 with r²=484: less means inside, equal means on boundary, and greater means outside. Here the result is outside.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0049/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 50. Three Positions Relative to a Circle Boundary

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -19) and radius r=23≥0. Classify P=(35, -19) as inside, on boundary, or outside.

View answer and explanationon boundary

Compare squared distance 529 with r²=529: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0050/)

Circle boundaries and zero-radius degeneracy · qf-circle-position · candidate





## 51. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=0≥0. Find its diameter length.

View answer and explanation0

The diameter length is 2r=0 in the same length unit, including zero at radius zero. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0051/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 52. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=1≥0. Find its diameter length.

View answer and explanation2

The diameter length is 2r=2 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0052/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 53. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=2≥0. Find its diameter length.

View answer and explanation4

The diameter length is 2r=4 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0053/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 54. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=3≥0. Find its diameter length.

View answer and explanation6

The diameter length is 2r=6 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0054/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 55. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=4≥0. Find its diameter length.

View answer and explanation8

The diameter length is 2r=8 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0055/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 56. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=5≥0. Find its diameter length.

View answer and explanation10

The diameter length is 2r=10 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0056/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 57. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=6≥0. Find its diameter length.

View answer and explanation12

The diameter length is 2r=12 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0057/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 58. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=7≥0. Find its diameter length.

View answer and explanation14

The diameter length is 2r=14 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0058/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 59. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=8≥0. Find its diameter length.

View answer and explanation16

The diameter length is 2r=16 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0059/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 60. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=9≥0. Find its diameter length.

View answer and explanation18

The diameter length is 2r=18 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0060/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 61. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=10≥0. Find its diameter length.

View answer and explanation20

The diameter length is 2r=20 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0061/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 62. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=11≥0. Find its diameter length.

View answer and explanation22

The diameter length is 2r=22 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0062/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 63. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=12≥0. Find its diameter length.

View answer and explanation24

The diameter length is 2r=24 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0063/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 64. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=13≥0. Find its diameter length.

View answer and explanation26

The diameter length is 2r=26 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0064/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 65. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=14≥0. Find its diameter length.

View answer and explanation28

The diameter length is 2r=28 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0065/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 66. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=15≥0. Find its diameter length.

View answer and explanation30

The diameter length is 2r=30 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0066/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 67. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=16≥0. Find its diameter length.

View answer and explanation32

The diameter length is 2r=32 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0067/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 68. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=17≥0. Find its diameter length.

View answer and explanation34

The diameter length is 2r=34 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0068/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 69. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=18≥0. Find its diameter length.

View answer and explanation36

The diameter length is 2r=36 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0069/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 70. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=19≥0. Find its diameter length.

View answer and explanation38

The diameter length is 2r=38 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0070/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 71. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=20≥0. Find its diameter length.

View answer and explanation40

The diameter length is 2r=40 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0071/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 72. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=21≥0. Find its diameter length.

View answer and explanation42

The diameter length is 2r=42 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0072/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 73. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=22≥0. Find its diameter length.

View answer and explanation44

The diameter length is 2r=44 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0073/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 74. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=23≥0. Find its diameter length.

View answer and explanation46

The diameter length is 2r=46 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0074/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 75. Diameter Length for a Nonnegative Radius

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=24≥0. Find its diameter length.

View answer and explanation48

The diameter length is 2r=48 in the same length unit, including zero at radius zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0075/)

Circle boundaries and zero-radius degeneracy · qf-circle-diameter · candidate





## 76. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, 0) and radius r=0≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(0, 0)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (0, 0), exactly on the boundary rather than inferred from a sketch. At zero radius the boundary degenerates to the singleton center, with no strictly inside points; all axial representations coincide.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0076/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 77. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-11, 2) and radius r=1≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-10, 2)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-10, 2), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0077/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 78. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-10, 1) and radius r=2≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-10, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-10, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0078/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 79. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-9, 0) and radius r=3≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-12, 0)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, 0), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0079/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 80. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-8, -1) and radius r=4≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-8, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-8, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0080/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 81. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-7, -2) and radius r=5≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-2, -2)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-2, -2), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0081/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 82. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -3) and radius r=6≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-6, -9)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-6, -9), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0082/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 83. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-5, -4) and radius r=7≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(2, -4)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (2, -4), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0083/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 84. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-4, -5) and radius r=8≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-4, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-4, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0084/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 85. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-3, -6) and radius r=9≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-12, -6)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -6), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0085/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 86. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-2, -7) and radius r=10≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-2, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-2, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0086/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 87. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-1, -8) and radius r=11≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(10, -8)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (10, -8), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0087/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 88. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(0, -9) and radius r=12≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(0, -21)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (0, -21), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0088/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 89. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(1, -10) and radius r=13≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(14, -10)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (14, -10), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0089/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 90. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(2, -11) and radius r=14≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(2, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (2, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0090/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 91. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(3, -12) and radius r=15≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-12, -12)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -12), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0091/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 92. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(4, -13) and radius r=16≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(4, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (4, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0092/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 93. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(5, -14) and radius r=17≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(22, -14)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (22, -14), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0093/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 94. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(6, -15) and radius r=18≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(6, -33)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (6, -33), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0094/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 95. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(7, -16) and radius r=19≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(26, -16)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (26, -16), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0095/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 96. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(8, -17) and radius r=20≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(8, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (8, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0096/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 97. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(9, -18) and radius r=21≥0. Change only its x-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(-12, -18)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (-12, -18), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0097/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 98. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(10, -19) and radius r=22≥0. Change only its y-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(10, 3)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (10, 3), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0098/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 99. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(11, -20) and radius r=23≥0. Change only its x-coordinate by (1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(34, -20)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (34, -20), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0099/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate





## 100. Specified Axial Point on a Circle

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(12, -21) and radius r=24≥0. Change only its y-coordinate by (-1)r. Give the resulting axial boundary point as an ordered pair.

View answer and explanation(12, -45)

Add the signed radius to the specified coordinate and leave the other unchanged. The point is (12, -45), exactly on the boundary rather than inferred from a sketch.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r05-circle-0100/)

Circle boundaries and zero-radius degeneracy · qf-circle-axial-point · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r05-circle-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261005-advance-20261006-r05-circle-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r05-circle-foundations/validation-report.json)