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Consider the ellipse x²/9 + y²/4 = 1 and the parabola y² = 2x, which intersect at points P (first quadrant) and Q (fourth quadrant). Tangents to the ellipse at P and Q meet the x-axis at R, and tangents to the parabola at P and Q meet the x-axis at S. Find the ratio of the areas of triangles PQS and PQR.

1. 1: 3
2. 1: 2
3. 2: 3
4. 3: 4

Correct answer: 1: 3

Solution

Solving gives x = 3/2, so P=(3/2, sqrt3), Q=(3/2, -sqrt3). The chord PQ is vertical. R (from ellipse tangents) is at x = 6 and S (from parabola tangents) is at x = -3/2; the ratio of distances from line x=3/2 gives areas 1:3.

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