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TMUA 2021 · Paper 1 · Question 20 of 20

TMUA 2021 Paper 1 Question 20

Coordinate geometry — Logarithms to a circle · arc length from the domain. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1Coordinate geometryLogarithms to a circle · arc length from the domain6 optionshard
Find the length of the curve with equation 2log10(x − y) = log10(2 − 2x) + log10(y + 5)
  1. A5
  2. B10
  3. C15
  4. D3π
  5. E9π
  6. F12π
Show the answer and worked solution
answer · D
  1. A5
  2. B10
  3. C15
  4. D3π
  5. E9π
  6. F12π
The logarithms force x − y > 0, x < 1 and y > −5; keep those aside and remove the logs to get (x−y)2 = (2−2x)(y+5). Expanding, the −2xy terms cancel and it reduces to x2 + 10x + y2 − 2y = 10, that is (x+5)2 + (y−1)2 = 36: a circle of radius 6 about (−5,  1), of total circumference 12π. Now impose x > y. The line y = x cuts the circle where 2x2 + 8x − 10 = 0, at (−5, −5) and (1, 1), a chord of length 6√2, which in a circle of radius 6 subtends 90∘ at the centre. The centre itself has x < y, so the arc that survives is the minor one — a quarter of the circle, of length 3π — and the other two conditions hold everywhere on it.