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Coordinate Geometry
Parabola, Focal Chord, Tangents
jee_adv_2026_mock_p1
Grade 12
Question:
Consider the parabola y^2 = 4ax. Let the focal chord PQ have parameters t1 and t2 with t1*t2 = -1. Which of the following is/are correct?
A. The length of the focal chord is a(t1 + t2)^2.
B. The minimum length of the focal chord is 4a.
C. If the focal chord is perpendicular to the axis, its length is 2a.
D. The product of slopes of tangents at P and Q is -1.
Step-by-Step Solution
Key Concept: Length of focal chord for parabola y^2 = 4ax is a(t1 - t2)^2 with t1*t2 = -1.
Step 1: Length of focal chord = a(t1 - t2)^2 = a[(t1 + t2)^2 + 4], so A is false. Step 2: Minimum occurs when t1 + t2 = 0, giving length = 4a, so B is true. Step 3: Focal chord perpendicular to axis is the latus rectum of length 4a, so C is false. Step 4: Slope of tangent at t is 1/t, product = 1/(t1*t2) = -1, so D is true.
Correct Answer:B, D
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