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TMUA 2022 · Paper 1 · Question 19 of 20

TMUA 2022 Paper 1 Question 19

Coordinate geometry — Circles · probability over a region of centres. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Coordinate geometryCircles · probability over a region of centres6 options
Circle C1 is defined as x2 + y2 = 25.

A second circle C2 has radius 4 and centre (a,  b) where −2 ≤ a ≤ 2  and  −3 ≤ b ≤ 3

If the centre of C2 is equally likely to be located anywhere within the given range, what is the probability that C2 intersects C1?

  1. A125
  2. B925
  3. C1625
  4. D6−π6
  5. E16−π24
  6. F24−π24
Show the answer and worked solution
answer · F
  1. A125
  2. B925
  3. C1625
  4. D6−π6
  5. E16−π24
  6. F24−π24
Two circles of radii 5 and 4 meet exactly when the distance d between centres satisfies 1 ≤ d ≤ 9. The centre lies in a 4 × 6 rectangle of area 24, where the largest possible d is √13 < 9, so the upper condition never bites. The lower one fails only when d < 1, that is inside the unit disc about the origin, which lies wholly within the rectangle and has area π. The probability is 24−π24.