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Let PQ be a focal chord of y² = 4ax (a > 0). The tangents at P and Q intersect at a point lying on the line y = 2x + a. If chord PQ subtends an angle theta at the vertex of the parabola, find tan(theta).

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

Correct answer: -2*sqrt(7)/3

Solution

The intersection of tangents lies on the directrix and on y = 2x + a, fixing t1+t2; using t1 t2 = -1 and the vertex-angle formula gives tan theta = -2 sqrt7/3 (this is the standard JEE 2013 answer).

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