I was doing an excercise, then I got stuck at ths question,
Find the locus of the mid point of the chord of contact of the tangents to the ellipse $x^2+4y^2 = 16$ which are at right angles. I tried to find the equation of the director circle and use it's properties but it didn't help. Then I thought to somehow get two relations between the mid point's coordinates and the slope of the chord of contact but I could find only one. $\alpha+4m\beta= 0$. Where $ m $ is the slope of chord of contact and $\alpha $ , $\beta$ are the cordinates of the mid point of the chord of contact.