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TMUA 2017 · Paper 1 · Question 6 of 20

TMUA 2017 Paper 1 Question 6

Coordinate geometry — Tangent to a circle · perpendicular distance from the centre. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 1Coordinate geometryTangent to a circle · perpendicular distance from the centre6 options
A tangent to the circle x2 + y2 = 144 passes through the point (20,  0) and crosses the positive y-axis.

What is the value of y at the point where the tangent meets the y-axis?

  1. A12
  2. B15
  3. C493
  4. D20
  5. E643
  6. F803
Show the answer and worked solution
answer · B
  1. A12
  2. B15
  3. C493
  4. D20
  5. E643
  6. F803
The circle is centred at the origin with radius 12, so a line is a tangent exactly when its distance from the origin is 12. Let the line meet the y-axis at (0,  k) with k > 0; through (20, 0) and (0, k) it is x20 + yk = 1, or kx + 20y − 20k = 0. Its distance from the origin is 20k√k2 + 400 = 12, so 400k2 = 144k2 + 57600 and k2 = 225. The positive root is k = 15.