Show that the points of intersection of the perpendicular tangents to an ellipse lies on a circle.
Let the equation of ellipse be
Any tangent to it is
Let the perpendicular tangents intersect at
Therefore,P lies on the tangent for some real m.
In another form,
is a quadratic equation in m. Let
its roots be m1,m2.Then,m1,m2 are the slopes of tangents from P to the ellipse.
Therefore,the point of intersection of perpendicular tangents to the ellipse lies on the circle