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TMUA 2019 · Paper 1 · Question 18 of 20

TMUA 2019 Paper 1 Question 18

Coordinate geometry — Shortest distance from a curve to a line · parallel tangent. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1Coordinate geometryShortest distance from a curve to a line · parallel tangent7 options
Find the shortest distance between the curve y = x2 + 4 and the line y = 2x − 2.
  1. A2
  2. B√5
  3. C6√55
  4. D3
  5. E5√53
  6. F5
  7. G6
Show the answer and worked solution
answer · B
  1. A2
  2. B√5
  3. C6√55
  4. D3
  5. E5√53
  6. F5
  7. G6
The closest point of the curve is the one whose tangent is parallel to the line. Since dydx = 2x = 2 gives x = 1, that point is (1,  5). Writing the line as 2x − y − 2 = 0, the perpendicular distance is |2(1) − 5 − 2|√22 + (−1)2 = 5√5 = √5. Measuring the vertical gap instead gives 5, which is option F and is the trap here.