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The locus of the midpoints of chords of the parabola y² = 2x that touch the circle x² + y² - 2x - 4 = 0 is (y² + 1 - x)² = lambda(1 + y²). Find lambda.

1. 5
2. 3
3. 4
4. 6

Correct answer: 5

Solution

Forming the midpoint chord of y² = 2x and requiring its distance from the circle centre (1,0) to equal radius sqrt 5 yields (y²+1-x)² = 5(1+y²), so lambda = 5.

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