The equations of tangent and normal to the ellipse at the point are and respectively.
Consider that the standard equation of ellipse with vertex at origin can be written as
Since the point lies on the given ellipse, it must satisfy equation (i). So we have
Now differentiating equation (i) on both sides with respect to , we have
If represents the slope of the tangent at the given point , then
The equation of a tangent at the given point is
This is the equation of the tangent to the given ellipse at .
The slope of the normal at is
The equation of the normal at the point is
This is the equation of the normal to the given ellipse at .