Question

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

• A. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{ex}{a}$
• B. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{ex}{a}$
• C. $x^2+y^2=a^2+b^2$
• D. None of these

Answer 1

The correct option is A $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{ex}{a}$

Let $(h,k)$ be the mid point of a focal chord. Then its equation is $T=S_1$

$\Rightarrow \frac{xh}{a^2}+\frac{ky}{b^2}-1=\frac{h^2}{a^2}+\frac{k^2}{b^2}-1$

Since it passes through $(ae,0)$

$\therefore \frac{hae}{a^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}$

$\therefore$ Locus of $(h,k)$ is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{xe}{a}$