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If the locus of the middle of point of contact of tangent drawn to the parabola $y^{2} = 8 x$ and the foot of perpendicular drawn from its focus to the tangent is a conic, then the length of latus rectum of this conic is

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answer is 9.

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Detailed Solution

Let the middle point of $P$ and $T$ be $(h, k) .$
Then, $2 h = a t^{2}$ and $2 k = 3 a t$
$∴ 2 h = a . \frac{4 k^{2}}{9 a^{2}}$
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The locus of $(h, k)$ is $2 y^{2} = 9 a x .$
As $a = 2,$ we get $y^{2} = 9 x .$

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