Question

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

• A. $\frac{9}{4}$
• B. $9$
• C. $18$
• D. $\frac{9}{2}$

Answer 1

The correct option is B $9$

Given equation of parabola is
$y^2=8x$
Let $P(2t^2,4t)$ be the point of contact of tangent to the parabola.
Equation of tangent at $P$ is
$ty=x+2t^2$
or $y=\frac{x}{t}+2t$ .....(i)
Equation of perpendicular line to tangent through focus $(2,0)$ is
$y=-t(x-2)$ .....(ii)
$\Rightarrow \frac{x}{t}+2t=-t(x-2)$
$\Rightarrow x=0$
$\Rightarrow y=2t$
Coordinates of $T$ are $(0,2t)$
Let $(h,k)$ be the middle point of $P$ and $T$
Then, $h=\frac{2t^2}{2}$ and $k=3t$
$\Rightarrow h={\left(\frac{k}{3}\right)}^2$
$y^2=9x$
Length of latus rectum is $9$.