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

Let $y=mx+\frac{1}{m}$ be any tangent to $y^2=4x$

Let $p(x_1,y_1)$ be the point of intersection $\Rightarrow$

$y_1=m{x}_1+\frac{1}{m}\Rightarrow x_1{m}^2-y_{1}m+1=0$ which is a quadratic equation in m.

If $m_1,m_2$ are the slopes of the two tangents then  $m_1,m_2$ are the roots of the equation since the tangents are perpendiculars $m_1,m_2=-1$

$\frac{1}{x_1}=-1\Rightarrow x_1+1=0$

$\therefore$locus of $p\left(x_1{y}_1\right)$ is $x+1=0\Rightarrow x=-1$