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TMUA 2022 · Paper 1 · Question 14 of 20

TMUA 2022 Paper 1 Question 14

Coordinate geometry — Circles · maximising a triangle on a fixed chord. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Coordinate geometryCircles · maximising a triangle on a fixed chord6 optionshard
A circle has centre O and radius 6.

P, Q and R are points on the circumference with angle POQ ≥ π2.

The area of the triangle POQ is 9√3.

What is the greatest possible area of triangle PRQ?

  1. A18 + 9√3
  2. B18√3
  3. C27 + 9√3
  4. D27√3
  5. E36 + 9√3
  6. F36√3
Show the answer and worked solution
answer · D
  1. A18 + 9√3
  2. B18√3
  3. C27 + 9√3
  4. D27√3
  5. E36 + 9√3
  6. F36√3
The area of POQ is 12(6)(6)sin(POQ) = 18sin(POQ) = 9√3, so sin(POQ) = √32 and, since the angle is at least π2, it is 2π3. The chord PQ then has length 2(6)sinπ3 = 6√3, and O sits 6cosπ3 = 3 from it. Placing R on the far arc puts it at most 6 + 3 = 9 from PQ, giving area 12(6√3)(9) = 27√3.