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TMUA 2021 · Paper 2 · Question 15 of 20

TMUA 2021 Paper 2 Question 15

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

TMUA 2021 · Paper 2Coordinate geometryCircles · tangency to an axis6 options
A circle has equation x2 + ax + y2 + by + c = 0 where a, b and c are non-zero real constants.

Which one of the following is a necessary and sufficient condition for the circle to be tangent to the y-axis?

  1. Aa2 = 4c
  2. Bb2 = 4c
  3. Ca2 = √a2+b24 − c
  4. Db2 = √a2+b24 − c
  5. E−a2 = √a2+b24 − c
  6. F−b2 = √a2+b24 − c
Show the answer and worked solution
answer · B
  1. Aa2 = 4c
  2. Bb2 = 4c
  3. Ca2 = √a2+b24 − c
  4. Db2 = √a2+b24 − c
  5. E−a2 = √a2+b24 − c
  6. F−b2 = √a2+b24 − c
Completing the square gives centre (−a2,  −b2) and radius r with r2 = a2+b24 − c. Tangency to the y-axis means the distance from the centre to that axis, namely |a2|, equals r. Squaring — which loses nothing here, as both sides are non-negative — gives a24 = a2+b24 − c, so c = b24, that is b2 = 4c. The options written with a square root each fix a sign for a or b, so they are sufficient at best, not equivalent.