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Coordinate Geometry
Ellipse, Tangents, Intercepts
jee_adv_2026_mock_p2
Grade 12
Question:
For the ellipse x^2/25 + y^2/9 = 1, let the tangent at a point P meet the axes at A and B. Which of the following is/are correct?
A. The product of intercepts is constant.
B. The minimum area of triangle OAB is 15.
C. The maximum area of triangle OAB is 25.
D. The locus of the midpoint of AB is 25/x^2 + 9/y^2 = 4.
Step-by-Step Solution
Key Concept: Tangent at (a cos θ, b sin θ) is x cos θ/a + y sin θ/b = 1.
Step 1: Tangent at (a cos θ, b sin θ) is x cos θ/a + y sin θ/b = 1. Step 2: Intercepts are a sec θ and b cosec θ. Step 3: Product = ab/(sin θ cos θ), not constant. Step 4: Area = ab/(2 sin θ cos θ) = ab/sin 2θ. Minimum area = ab = 15 at θ = π/4. Maximum is infinite. Step 5: Midpoint is (a sec θ/2, b cosec θ/2). Locus: 25/x^2 + 9/y^2 = 4. So B and D are true, A and C are false.
Correct Answer:B, D
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