Geometry and TrigonometryVery hardExpanded form of a circle equation
Geometry and Trigonometry practice question
Which of the following equations represents the circle with centre (3, −1) and radius 5 in the xy-plane?
- A. x² + y² − 6x + 2y − 15 = 0Correct
- B. x² + y² + 6x − 2y − 15 = 0
- C. x² + y² − 6x + 2y + 25 = 0
- D. x² + y² − 3x + y − 25 = 0
Answer: A. x² + y² − 6x + 2y − 15 = 0
(x − 3)² + (y + 1)² = 25 expands to x² − 6x + 9 + y² + 2y + 1 = 25, i.e. x² + y² − 6x + 2y − 15 = 0.
Why the other answers are wrong
- B. x² + y² + 6x − 2y − 15 = 0
- +6x − 2y comes from centre (−3, 1).
- C. x² + y² − 6x + 2y + 25 = 0
- +25 forgets to bring the 9 + 1 across and subtract; the constant should be 10 − 25 = −15.
- D. x² + y² − 3x + y − 25 = 0
- −3x + y halves the linear coefficients; expanding (x − 3)² gives −6x.
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