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Perpendicular Distance of a Point from a Line

The locus of ...

Question

The locus of point of intersection of two tangents to $y^{2} = 4 a x$ at $t$ and $2 t$ on the parabola is

2 y^{2} = 9 a x

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4 y^{2} = 9 a x

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3 y^{2} = 4 a x

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3 y^{2} = 8 a x

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Solution

The correct option is A $2 y^{2} = 9 a x$
Point of intersection of two tangents to $y^{2} = 4 a x$ at $t$ and $2 t$ is $(2 a t^{2}, 3 a t)$
Let $(h, k)$ be the locus
So, $3 a t = k \Rightarrow t = \frac{k}{3 a}$
and $h = 2 a t^{2} \Rightarrow h = 2 a {(\frac{k}{3 a})}^{2}$
$\Rightarrow 2 k^{2} = 9 a h$
$∴ 2 y^{2} = 9 a x$

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Perpendicular Distance of a Point from a Line

Standard XII Mathematics

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The locus of point of intersection of two tangents to y2=4ax at t and 2t on the parabola is

The correct option is A 2y2=9axPoint of intersection of two tangents to y2=4ax at t and 2t is (2at2,3at) Let (h,k) be the locus So, 3at=k⇒t=k3a and h=2at2⇒h=2a(k3a)2 ⇒2k2=9ah ∴2y2=9ax

The locus of point of intersection of two tangents to $y^{2} = 4 a x$ at $t$ and $2 t$ on the parabola is