Question

# If two tangents drawn from a point P to the parabola ${y^2} = 4x$ are at right angles, then the locus of P isA. $x = 1$B. $2x + 1 = 0$C. $x = - 1$D. $2x - 1 = 0$

We have given the equation of parabola as ${y^2} = 4x$. Let’s consider it as an equation ……..(1).
Also, the standard equation to the parabola is ${y^2} = 4ax$ and when we shift the origin to the point $(h,k)$ then this equation will become ${(y - k)^2} = 4p(x - h)$. It has focus $(h + p,k)$ and the directrix is $x = h - p$.
From equation (1), we get, $h = 0,k = 0,p = 1$. So, the directrix of equation (1) is $x = 0 - 1 = - 1$. Hence, required locus is $x = - 1$.