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TMUA 2021 · Paper 1 · Question 8 of 20

TMUA 2021 Paper 1 Question 8

Algebra and functions — Discriminant · two tangent lines to one parabola. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1Algebra and functionsDiscriminant · two tangent lines to one parabola6 options
The line y = 2x + 3 meets the curve y = x2 + bx + c at exactly one point.

The line y = 4x − 2 also meets the curve y = x2 + bx + c at exactly one point.

What is the value of b − c?

  1. A−9
  2. B−5.5
  3. C−1
  4. D5
  5. E6
  6. F14
Show the answer and worked solution
answer · A
  1. A−9
  2. B−5.5
  3. C−1
  4. D5
  5. E6
  6. F14
Meeting at exactly one point means the quadratic formed by equating them has a repeated root, so its discriminant is zero. From the first line, x2 + (b−2)x + (c−3) = 0 gives (b−2)2 = 4(c−3). From the second, x2 + (b−4)x + (c+2) = 0 gives (b−4)2 = 4(c+2). Subtracting eliminates c: (b−2)2 − (b−4)2 = 4b − 12 = −20, so b = −2. Then (b−2)2 = 16 = 4(c−3) gives c = 7, and b − c = −9.