Question

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

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

Answer 1

Answer: (A)

Let $(h, k)$ be the mid point of a focal chord.
Then its equation is $T=S_{1}$
i.e., $\frac{x h}{a^{2}}+\frac{k y}{b^{2}}-1=\frac{h^{2}}{a^{2}}+\frac{k^{2}}{b^{2}}-1$
Since it passes through $(a e, 0)$
$\therefore \frac{h a e}{a^{2}}=\frac{h^{2}}{a^{2}}+\frac{k^{2}}{b^{2}}$
$\therefore $ Locus of $(h, k)$ is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\frac{x e}{a}$