# locus of point of intersection of tangents to a parabola

A line intersects the ellipse $$\frac{x^2}{4a^2}+\frac{y^2}{a^2}=1$$ at $A$ and $B$ and the parabola $$y^2=4a(x+2a)$$ at $C$ and $D$.The line segment $AB$ subtends a right angle at the centre of the ellipse.Then,the question is to find out the locus of the point of intersection of tangents to the parabola at $C$ and $D$.

Let the line be $y=mx+c$ .Since it intersects the ellipse in two points I substituted it in the equation of ellipse and hence got the x coordinates of point of intersection.I then tried to homogeneralise the equation of elipse and equating the coefficients of $x^2$ and $y^2$ to zero.I couldnot proceed after this.Moreover this approach is on the longer side.Is there any shorter approach possible.This question came in an objective examination in which time was restricted.

Thanks.