Geometry and TrigonometryVery hardCentre from the general form of a circle
Geometry and Trigonometry practice question
The equation x² + y² + 4x − 6y − 3 = 0 represents a circle in the xy-plane. What are the coordinates of the centre of the circle?
- A. (−2, 3)Correct
- B. (2, −3)
- C. (−4, 6)
- D. (4, −6)
Answer: A. (−2, 3)
Complete the square: (x² + 4x + 4) + (y² − 6y + 9) = 3 + 4 + 9, so (x + 2)² + (y − 3)² = 16. The centre is (−2, 3).
Why the other answers are wrong
- B. (2, −3)
- (2, −3) reverses the signs. In (x + 2)², the centre's x-coordinate is −2.
- C. (−4, 6)
- (−4, 6) uses the coefficients themselves; the centre's coordinates are HALF the coefficients, with signs flipped.
- D. (4, −6)
- (4, −6) uses the coefficients with the wrong signs and without halving.
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