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TMUA 2020 · Paper 1 · Question 16 of 20

TMUA 2020 Paper 1 Question 16

Coordinate geometry — Circles · common tangent to two equal circles. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1Coordinate geometryCircles · common tangent to two equal circles8 options
The circle C1 has equation (x+2)2 + (y−1)2 = 3.

The circle C2 has equation (x−4)2 + (y−1)2 = 3.

The straight line l is a tangent to both C1 and C2 and has positive gradient.

The acute angle between l and the x-axis is θ.

Find the value of tanθ.

  1. A12
  2. B2
  3. C√22
  4. D√2
  5. E√62
  6. F√63
  7. G√33
  8. H√3
Show the answer and worked solution
answer · C
  1. A12
  2. B2
  3. C√22
  4. D√2
  5. E√62
  6. F√63
  7. G√33
  8. H√3
The centres are (−2, 1) and (4, 1), six apart, and both radii are √3. Because the radii are equal, the common tangents are either parallel to the line of centres — horizontal, so zero gradient — or pass through the midpoint (1, 1). A tangent with positive gradient must be one of the latter, so write it as y − 1 = m(x−1), that is mx − y + 1 − m = 0. Its distance from (−2, 1) must be √3: |−2m − 1 + 1 − m|√m2+1 = 3|m|√m2+1 = √3, so 9m2 = 3m2 + 3 and m2 = 12. Taking the positive root, tanθ = 1√2 = √22.