Question

Find the locus of the middle points of chords of an ellipse whose distance from the centre is the constant length $c$.

Answer 1

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

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

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

$\frac{x{x}^{\prime }}{a^2}+\frac{y{y}^{\prime }}{b^2}=\frac{x^{\prime 2}}{a^2}+\frac{y^{\prime 2}}{b^2}\frac{xh}{a^2}+\frac{yk}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}{b}^{2}xh+a^{2}ky=b^2{h}^2+a^2{k}^2{b}^{2}hx+a^{2}ky-b^2{h}^2-a^2{k}^2=0$

Distance from centre = $c$

$\left|\frac{0\left(b^{2}h\right)+0\left(a^{2}k\right)-b^2{h}^2-a^2{k}^2}{\sqrt{{\left(b^{2}h\right)}^2+{\left(a^{2}k\right)}^2}}\right|=c$

Squaring both sides

${(b^2{h}^2+a^2{k}^2)}^2=c^2({\left(b^{2}h\right)}^2+{\left(a^{2}k\right)}^2)$

So the required locus is

${(b^2{x}^2+a^2{y}^2)}^2=c^2({\left(b^{2}x\right)}^2+{\left(a^{2}y\right)}^2){(b^2{x}^2+a^2{y}^2)}^2=c^2(b^4{x}^2+a^4{y}^2)$