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TMUA 2022 · Paper 1 · Question 2 of 20

TMUA 2022 Paper 1 Question 2

Coordinate geometry — Circles · when the equation describes one. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Coordinate geometryCircles · when the equation describes one6 options
Find the complete set of values of p for which the equation x2 − 2px + y2 − 6y − p2 + 8p + 9 = 0 describes a circle in the xy-plane.
  1. Ap < −94
  2. B0 < p < 4
  3. C−1 < p < 9
  4. Dp < 0 or p > 4
  5. Ep < −1 or p > 9
  6. Fall real values of p
Show the answer and worked solution
answer · D
  1. Ap < −94
  2. B0 < p < 4
  3. C−1 < p < 9
  4. Dp < 0 or p > 4
  5. Ep < −1 or p > 9
  6. Fall real values of p
Completing both squares gives (x−p)2 − p2 + (y−3)2 − 9 − p2 + 8p + 9 = 0, that is (x−p)2 + (y−3)2 = 2p2 − 8p. This is a genuine circle only when the right-hand side is strictly positive: 2p(p−4) > 0, so p < 0 or p > 4.