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The lines y = m1*x + c1 and y = m2*x + c2 (with m1 not equal to m2) are the two common tangents of the circle x² + y² = 2 and the parabola y² = x. Find the value of 8*|m1*m2|.

1. -5 + 6*sqrt(2)
2. 3 + 4*sqrt(2)
3. -4 + 3*sqrt(2)
4. 7 + 6*sqrt(2)

Correct answer: -5 + 6*sqrt(2)

Solution

For the parabola y² = x (4a = 1, so a = 1/4), the tangent is y = mx + 1/(4m); imposing tangency to the circle gives a quadratic in m² whose root product leads to 8|m1 m2| = -5 + 6*sqrt(2).

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