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TMUA 2023 · Paper 1 · Question 13 of 20

TMUA 2023 Paper 1 Question 13

Coordinate geometry — Circles · greatest distance between two points. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Coordinate geometryCircles · greatest distance between two points7 options
Point P lies on the circle with equation (x−2)2 + (y−1)2 = 16.

Point Q lies on the circle with equation (x−4)2 + (y+5)2 = 16.

What is the maximum possible length of PQ?

  1. A10
  2. B14
  3. C16
  4. D2√34
  5. E10√2
  6. F8 + 2√10
  7. G16 + 2√6
Show the answer and worked solution
answer · F
  1. A10
  2. B14
  3. C16
  4. D2√34
  5. E10√2
  6. F8 + 2√10
  7. G16 + 2√6
The centres are (2, 1) and (4, −5), a distance √22 + 62 = √40 = 2√10 apart, and both radii are 4. The greatest separation puts P and Q at opposite ends of the line through the centres, giving 2√10 + 4 + 4 = 8 + 2√10.