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Product of Abscissae of Tangents on Tangent at Vertex | JEE

JEE Maths question with a full step-by-step solution.

Question
If the tangents at the extremities of a focal chord of the parabola x2=4ayx^2 = 4ay meet the tangent at the vertex at points whose abscissae are x1x_1 and x2x_2, then x1x2x_1 x_2 is:
Aa2a^2
Ba2−1a^2 - 1
Ca2+1a^2 + 1
D−a2-a^2correct
Solution
Step 1: Tangent at (2at,at2)(2at, at^2) on x2=4ayx^2 = 4ay is tx=y+at2tx = y + at^2; at the vertex tangent y=0y = 0 it gives x1=atx_1 = at. Step 2: The other end has parameter −1t-\dfrac{1}{t} (focal chord, t1t2=−1t_1 t_2 = -1), so x2=−atx_2 = -\dfrac{a}{t}. Hence
x1x2=(at)(−at)=−a2.x_1 x_2 = (at)\left(-\frac{a}{t}\right) = -a^2.
 Correct answer: (4)
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