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WMA11/01 Jun 2024, Question 4: Lines Meeting Curves: the Discriminant

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 6 marks

The question, in short

Two curves with constant k are given; they intersect at exactly one point P. Show a quadratic in k and use it to find P's coordinates.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Set the two curve equations equal, form a quadratic in x, and use the single-intersection condition to show k² + 26k + 169 = 0.

  1. 1x² + kx − 9 = −3x² − 5x + k
  2. 24x² + (k + 5)x − 9 − k = 0
  3. 3One meeting point, so b² − 4ac = 0: (k + 5)² − 4(4)(−9 − k) = 0
  4. 4k² + 10k + 25 + 144 + 16k = 0
  5. 5k² + 26k + 169 = 0

AnswerShown: 4x² + (k + 5)x − 9 − k = 0 has one root, so (k + 5)² − 16(−9 − k) = 0, giving k² + 26k + 169 = 0

Part (b) · 3 marks

Solve the equation for k, then substitute to find the coordinates of the single intersection point P.

  1. 1(k + 13)² = 0, so k = −13
  2. 24x² − 8x + 4 = 0, so x² − 2x + 1 = 0
  3. 3(x − 1)² = 0, so x = 1
  4. 4y = 1 − 13 − 9 = −21, so P = (1, −21)

AnswerP = (1, −21)

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