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Perpendicular Distance of a Point from a Line

If the point ...

Question

If the point of intersection of tangents at $t_{1}$ and $t_{2}$ to the parabola $y^{2} = 8 x$ lies on the line $x + y + 2 = 0$ , then value of $(1 + t_{1}) (1 + t_{2})$ is

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Solution

The point of intersection of tangents at $t_{1}$ and $t_{2}$ is $(a t_{1} t_{2}, a (t_{1} + t_{2}))$
Here, $a = 2$
$∴$ The point is $(2 t_{1} t_{2}, 2 (t_{1} + t_{2}))$

Now this point lies on $x + y + 2 = 0$
$\Rightarrow 2 t_{1} t_{2} + 2 (t_{1} + t_{2}) + 2 = 0 \Rightarrow t_{1} t_{2} + (t_{1} + t_{2}) + 1 = 0 ∴ (1 + t_{1}) (1 + t_{2}) = 0$

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Perpendicular Distance of a Point from a Line

Standard XII Mathematics

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