WMA11/01 Oct 2019, Question 6: Lines Meeting Curves: the Discriminant
Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 8 marks
The question, in short
A curve has equation y = 4/x + k with k positive. Draw the curve, give its horizontal asymptote and x-axis crossing, then use a tangent line y = 10 − 2x to find k.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 3 marks
Draw the graph of y = 4/x + k, giving the horizontal asymptote's equation and the coordinates where it meets the x-axis.
- 1The graph of y = 4/x is translated k units up
- 2Horizontal asymptote: y = k; vertical asymptote: x = 0
- 3Crosses the x-axis where 4/x + k = 0, so x = −4/k: point (−4/k, 0)
- 4Right branch lies above y = k; left branch lies below y = k and crosses the negative x-axis
AnswerTwo-branch curve y = 4/x shifted up by k: one branch in the first quadrant, the other in the second/third quadrants crossing the negative x-axis at (−4/k, 0); horizontal asymptote y = k, vertical asymptote x = 0 (the y-axis)
Part (b) · 5 marks
Given that y = 10 − 2x touches the curve, determine the possible values of k.
- 110 − 2x = 4/x + k; multiply by x: 10x − 2x² = 4 + kx
- 22x² + (k − 10)x + 4 = 0
- 3Tangent means equal roots: (k − 10)² − 4 × 2 × 4 = 0
- 4(k − 10)² = 32, so k − 10 = ±4√2
- 5k = 10 ± 4√2 (both positive)
Answerk = 10 ± 4√2
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