Question

Find the locus of the middle points of chords of an ellipse which pass through the given point $(h,k)$.

Answer 1

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Let $(p,q)$ be the mid point of the chord

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

$\frac{x{x}^{\prime }}{a^2}+\frac{y{y}^{\prime }}{b^2}=\frac{x^{\prime 2}}{a^2}+\frac{y^{\prime 2}}{b^2}\frac{xp}{a^2}+\frac{yq}{b^2}=\frac{p^2}{a^2}+\frac{q^2}{b^2}$

It passes through $(h,k)$

$\frac{hp}{a^2}+\frac{kq}{b^2}=\frac{p^2}{a^2}+\frac{q^2}{b^2}{b}^{2}hp+a^{2}kq=b^2{p}^2+a^2{q}^2{b^2{p}^2-b^{2}hp+a}^2{q}^2-a^{2}kq=0$

Replacing $p$ by $x$ and $q$ by $y$

${b^2{x}^2-b^{2}hx+a}^2{y}^2-a^{2}ky=0b^{2}x(x-h)+a^{2}y(y-k)=0$

is the required equation of locus.