Question

Find the locus of the middle points of chords of an ellipse which subtend a right angle at the centre.

Answer 1

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\mathrm{1.....}\left(i\right)$

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

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

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

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

Making the equation of the homogenous with the equation of the chord

$\frac{(\frac{xh}{a^2}+\frac{yk}{b^2})}{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}=1$

Substituing in $\left(i\right)$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}={\left(\frac{(\frac{xh}{a^2}+\frac{yk}{b^2})}{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}\right)}^2{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2\frac{x^2}{a^2}+{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2\frac{y^2}{b^2}={(\frac{xh}{a^2}+\frac{yk}{b^2})}^2{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2\frac{x^2}{a^2}+{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2\frac{y^2}{b^2}=\frac{h^2{x}^2}{a^4}+\frac{k^2{y}^2}{b^4}+\frac{2hk}{a^2{b}^2}xy({(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2-\frac{h^2}{a^2})\frac{x^2}{a^2}+({(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2-\frac{k^2}{b^2})\frac{y^2}{b^2}-\frac{2hk}{a^2{b}^2}xy=0$

The pair of line is perpendicular

$\therefore a+b=0\frac{1}{a^2}({(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2-\frac{h^2}{a^2})+\frac{1}{b^2}({(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2-\frac{k^2}{b^2})=0\frac{1}{a^2}{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2-\frac{h^2}{a^4}+\frac{1}{b^2}{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2-\frac{k^2}{b^4}=0\frac{1}{a^2}{\left(\frac{b^2{h}^2+a^2{k}^2}{a^2{b}^2}\right)}^2-\frac{h^2}{a^4}+\frac{1}{b^2}{\left(\frac{b^2{h}^2+a^2{k}^2}{a^2{b}^2}\right)}^2-\frac{k^2}{b^4}=0{\left(\frac{b^2{h}^2+a^2{k}^2}{a^2{b}^2}\right)}^2(\frac{1}{a^2}+\frac{1}{b^2})=\frac{h^2}{a^4}+\frac{k^2}{b^4}{\left(\frac{b^2{h}^2+a^2{k}^2}{a^2{b}^2}\right)}^2\frac{a^2+b^2}{a^2{b}^2}=\frac{b^4{h}^2+a^4{k}^2}{a^4{b}^4}(a^2+b^2){(b^2{h}^2+a^2{k}^2)}^2=a^2{b}^2(b^4{h}^2+a^4{k}^2)$

Replacing $h$ by $x$ and $k$ by $y$

$(a^2+b^2){(b^2{x}^2+a^2{y}^2)}^2=a^2{b}^2(b^4{x}^2+a^4{y}^2)$

is the required equation of locus.