Question

Find the locus of the middle points of chords of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which are drawn through the positive end of the minor axis.

Answer 1

Let the ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Let the mid point of chord of contact be $(h,k)$

Equation of chord when mid point is given is $T=S^{\prime }$

$\frac{x{x}^{\prime }}{a^2}+\frac{y{y}^{\prime }}{b^2}=\frac{x^{\prime 2}}{a^2}+\frac{y^{\prime 2}}{b^2}\frac{xh}{a^2}+\frac{yk}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}$

It passes through $(0,b)$

$0+\frac{bk}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}\frac{h^2}{a^2}+\frac{k^2}{b^2}=\frac{k}{b}$

So the required locus is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{y}{b}$