Question

If two tangents drawn from a point $P$ to the parabola $y^2=4x$ are at right angles, then the locus of $P$ is

• A. $x=1$
• B. $2x+1=0$
• C. $x=-1$
• D. $2x-1=0$

Answer 1

The correct option is C $x=-1$

Given equation of parabola is $y^2=4x$ ...... $\left(i\right)$

We know that the locus of point $P$ from which two perpendicular tangents are drawn to the parabola is the directrix of the parabola.

The standard equation of parabola $(y-k{)}^2=4p(x-h)$ has focus $(h+p,k)$ and the directrix is $x=h-p$

From $\left(i\right)$, we get

$h=0,k=0,p=1$

$\Rightarrow$ Directrix of $\left(i\right)$ is $x=0-1=-1$

Hence, required locus is $x=-1$.