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

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 $\left(-2, 0\right)\Rightarrow \left(-k, 0\right) =\left(-2, 0\right) \Rightarrow k=2$