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TMUA 2019 · Paper 1 · Question 6 of 20

TMUA 2019 Paper 1 Question 6

Coordinate geometry — Circles · internal and external tangency. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1Coordinate geometryCircles · internal and external tangency5 options
The circles with equations (x+4)2 + (y+1)2 = 64 and (x−8)2 + (y−4)2 = r2,    where  r > 0 have exactly one point in common.

Find the difference between the two possible values of r.

  1. A4
  2. B10
  3. C16
  4. D26
  5. E50
Show the answer and worked solution
answer · C
  1. A4
  2. B10
  3. C16
  4. D26
  5. E50
The centres are (−4, −1) and (8, 4), a distance √122 + 52 = 13 apart, and the first radius is 8. Two circles touch at exactly one point either externally, when the centre distance equals the sum of the radii, or internally, when it equals the difference. Externally: 8 + r = 13, so r = 5. Internally: |r − 8| = 13, and since r > 0 this gives r = 21 (the small circle would have to swallow the large one). The difference is 21 − 5 = 16.