Question

The locus of mid-points of a focal chord 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

Answer: (A)

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

Answer 2

Let $(h,k)$ be the mid point of a focal chord. Then its equation is $S_{1} = T$or $\frac{hx}{a^{2}} + \frac{ky}{b^{2}} = \frac{h^{2}}{a^{2}} + \frac{k^{2}}{b^{2}}$. This passes through $(ae,0)$, ∴$\frac{hae}{a^{2}} = \frac{h^{2}}{a^{2}} + \frac{k^{2}}{b^{2}}$. So, locus of $(h,k)$ is $\frac{xe}{a} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}$