Question

The equation of a tangent to the parabola $y^2=8x$ is $y=x+2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is $\left(-k,0\right)$, then the value of $k$ is

Answer 1

Answer: 2

Parabola $y^2=8x$

We know that the locus of point of intersection of two perpendicular tangents to a parabola is its directrix. Point must be on the directrix of parabola
$\therefore$ equation of directrix $x+2=0$
$\Rightarrow x=-2$
Then the point is $\left(-2,0\right)\Rightarrow \left(-k,0\right)=\left(-2,0\right)\Rightarrow k=2$