Question

Find the locus of the mid point of focal chords of the ellipse$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Answer 1

Let P=(h, k) be the mid-point
$\therefore$ equation of chord whose mid-point is given $\frac{xh}{a^2}+\frac{yk}{b^2}-1=\frac{h^2}{a^2}+\frac{k^2}{b^2}-1$Since it is a focal chord,
$\therefore$ it passes through focus, either (ae, 0) or (-ae, 0) If it passes through (ae, 0)
$\therefore$ locus is $\frac{ex}{a}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ If it passes through (-ae, 0)
$\therefore$ locus is $-\frac{ex}{a}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$