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TMUA 2022 · Paper 2 · Question 4 of 20

TMUA 2022 Paper 2 Question 4

Coordinate geometry — Circles · what determines a distance. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Coordinate geometryCircles · what determines a distance5 options
The point P has coordinates (p,  q), and the equation of a circle is x2 + 2fx + y2 + 2gy + h = 0 where f, g, h, p and q are all real constants.

Let L be the distance between the centre of the circle and the point P.

Which one of the following is sufficient on its own to be able to calculate L?

  1. Athe values of f, g and h
  2. Bthe values of f, g, p and q
  3. Cthe values of f, h, p and q
  4. Dthe values of g, h, p and q
  5. Enone of the options A–D is sufficient on its own
Show the answer and worked solution
answer · B
  1. Athe values of f, g and h
  2. Bthe values of f, g, p and q
  3. Cthe values of f, h, p and q
  4. Dthe values of g, h, p and q
  5. Enone of the options A–D is sufficient on its own
Completing the square puts the circle in the form (x+f)2 + (y+g)2 = f2 + g2 − h, so its centre is (−f,  −g). The constant h affects only the radius, which the distance does not need. So L = √(p+f)2 + (q+g)2 requires exactly f, g, p and q.