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Pair of Tangents from an External Point

If the tangen...

Question

If the tangents are drawn at $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ on the parabola $y^{2} = 4 a x$ intersect on axis of the parabola, then

t_{1} t_{2} = 2

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t_{1} t_{2} = - 4

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t_{1} t_{2} = - 1

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t_{1} = - t_{2}

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Solution

The correct option is D $t_{1} = - t_{2}$
For $y^{2} = 4 a x$ , tangent at $(a t_{1}^{2}, 2 a t_{1})$ is $y t_{1} = x + a t_{1}^{2} \dots (1)$
For $y^{2} = 4 a x$ , tangent at $(a t_{2}^{2}, 2 a t_{2})$ is $y t_{2} = x + a t_{2}^{2} \dots (2)$
Intersection point is $(a t_{1} t_{2}, a (t_{1} + t_{2}))$
Now intersection of $(1)$ and $(2)$ is on $x -$ axis that is $(x_{0}, 0)$
$a (t_{1} + t_{2}) = 0 \Rightarrow t_{1} = - t_{2}$

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Pair of Tangents from an External Point

Standard XII Mathematics

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