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TMUA 2018 · Paper 1 · Question 3 of 20

TMUA 2018 Paper 1 Question 3

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

TMUA 2018 · Paper 1Coordinate geometryCircles · shortest distance between two circles5 options
Find the shortest distance between the two circles with equations: (x + 2)2 + (y − 3)2 = 18 (x − 7)2 + (y + 6)2 = 2
  1. A0
  2. B4
  3. C16
  4. D2√2
  5. E5√2
Show the answer and worked solution
answer · E
  1. A0
  2. B4
  3. C16
  4. D2√2
  5. E5√2
The centres are (−2,  3) and (7,  −6), with radii √18 = 3√2 and √2. The centres are √92 + 92 = √162 = 9√2 apart, which is more than the sum of the radii, so the circles are separate. The nearest points lie on the line joining the centres, so the gap is 9√2 − 3√2 − √2 = 5√2.