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# 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. $2x - 1 = 0$

Last updated date: 16th Mar 2023
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Hint: Use the information that the locus of point p from which two perpendicular tangents are drawn to the parabola is the directrix of the parabola. So, essentially, we need to find the directrix of the given parabola.

We have given the equation of parabola as ${y^2} = 4x$. Let’s consider it as an equation ……..(1).
We know that the locus of point p from which two perpendicular tangents are drawn to the parabola is the directrix of the parabola.
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 $x = - 1$.

Note: Locus, as the word says, is the path of a point under given conditions. Here we observed that the path of the point will be the directrix of the given parabola then we solved the problem.