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R0004-02904 · q-20261005-advance-20261006-r05-circle-0079
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.
G001 — 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.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
orthonormal equal-length units;r=3≥0
After
C={P:distance(P,O)=r} is a boundary,not a filled diskConditions: none
The circle is the boundary at distance exactly r from the center,not the filled disk.For r=0 use the degenerate singleton{O},with no inside points and diameter length0.
Rule: circle_boundary_convention · Uses: G001
Before
radius squared := square(3)
After
radius squared = 9
Conditions: none
The square of the nonnegative radius is the distance-comparison threshold.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
signed radial displacement := mul(-1, 3)
After
signed radial displacement = -3
Conditions: none
Multiply the radius by the specified direction sign;only the selected coordinate changes.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
point coordinate0 := sum(-9, -3)
After
point coordinate0 = -12
Conditions: none
Add displacement to the center;the unselected coordinate adds0 and is retained.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
point coordinate1 := sum(0, 0)
After
point coordinate1 = 0
Conditions: none
Add displacement to the center;the unselected coordinate adds0 and is retained.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
coordinate difference0 := sub(-12, -9)
After
coordinate difference0 = -3
Conditions: none
Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
coordinate square0 := square(-3)
After
coordinate square0 = 9
Conditions: none
Square this coordinate difference;a negative difference still contributes a nonnegative square.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
coordinate difference1 := sub(0, 0)
After
coordinate difference1 = 0
Conditions: none
Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
coordinate square1 := square(0)
After
coordinate square1 = 0
Conditions: none
Square this coordinate difference;a negative difference still contributes a nonnegative square.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
squared distance := sum(9, 0)
After
squared distance = 9
Conditions: none
In an orthonormal equal-unit frame,the Pythagorean relation adds the squared differences.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
squared distance=9;r²=9
After
(-12, 0)
Conditions: none
Substitute back to check circle membership;when r=0 every specified axis and direction yields the center.
Rule: axial_point_membership_audit · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute squared distance with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute coordinate square1 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute coordinate difference1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute coordinate square0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute coordinate difference0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute point coordinate1 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute point coordinate0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute signed radial displacement with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute radius squared with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(-12, 0)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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