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Geometry and TrigonometryEasyEquation of a circle in the xy-plane

Geometry and Trigonometry practice question

In the xy-plane, a circle has center (2, −5) and radius 3. Which equation represents this circle?

  1. A. (x − 2)² + (y + 5)² = 9Correct
  2. B. (x + 2)² + (y − 5)² = 9
  3. C. (x − 2)² + (y + 5)² = 3
  4. D. (x + 2)² + (y − 5)² = 3

Answer: A. (x − 2)² + (y + 5)² = 9

The standard form is (x − h)² + (y − k)² = r² with center (h, k) and radius r. Here h = 2, k = −5, r = 3, so the equation is (x − 2)² + (y − (−5))² = 3², or (x − 2)² + (y + 5)² = 9. (B) reverses both signs of the center coordinates, reading the center as (−2, 5). (C) uses r instead of r² on the right side, forgetting to square the radius. (D) makes both errors: wrong signs for the center and an unsquared radius.

Why the other answers are wrong

B. (x + 2)² + (y − 5)² = 9
You flipped the signs of the center coordinates the wrong way. In (x − h)² + (y − k)², h = 2 gives (x − 2)² and k = −5 gives (y + 5)².
C. (x − 2)² + (y + 5)² = 3
You put the radius itself on the right side instead of its square; it must be r² = 3² = 9.
D. (x + 2)² + (y − 5)² = 3
Two things went wrong here: the center signs are reversed (this equation has center (−2, 5)) and the right side should be r² = 9, not 3.

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