Find the locus of the point of intersection of the normals at the end of the focal chord of the parabola `y^2=4a xdot`

Options

A. tangent at the vertex
B. its derectrix
C. its latusrectum
D. a parabola

Answer

B

Explanation

Let P (h, k) be the point of intersection of tengents at points `Q(at_(1)^(2), 2at_(1))" and R"(at_(2)^(2), 2at_(2))` on the parabola `y^(2)=4ax`. Then, `h=at_(1)t_(2)" and "k=a (t_(1)+t_(2))`
The equation of tangents at Q and R are
`t_(1)t=x+at_(1)^(2)" and "t_(2)y=x+at_(2)^(2)`
These two will be perpendicualar, if `1/t_(1)xx1/t_(2)=-1rArrt_(1)t_(2)=-1`
Putting `t_(1)t_(2)=-1" in "h=at_(1)t_(2)," we get h"=-a`
So, the locus of (h, k) is x= -a, which is the directrix of the parabola.