Question

Find the locus (1) of the middle points, and (2) of the poles, of normal chords of the ellipse.

Answer 1

The chord, whose middle point is (h, k), is parallel to the polar of (h, k) , and is therefore

If this be a normal, it must be the same as

The equation to the required locus is therefore

Again, if (x₁, y₁) be the pole of the normal chord (2), the latter equation must be equivalent to the equation

Comparing (2) and (3), we have

and hence the required locus is