Question

If the locus of the middle points of chords of an ellipse $\frac{x^2}{3}+\frac{y^2}{4}=1$ passing through $(2,0)$ is another ellipse $A$, then the length of latus rectum of the ellipse $A$ is

• A. $\frac{8}{3}$
• B. $\sqrt{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{3}{8}$

Answer 1

The correct option is B $\sqrt{3}$

Let middle point of chord be $P(h,k)$
So equation of chord with P as mid point is given by,
$T=S_1\Rightarrow \frac{xh}{3}+\frac{yk}{4}=\frac{h^2}{3}+\frac{k^2}{4}$
Given chord passing through $(2,0).$
$\Rightarrow \frac{2h}{3}=\frac{h^2}{3}+\frac{k^2}{4}$

$\Rightarrow \frac{(h-1{)}^2}{1}+\frac{k^2}{4/3}=1$
Hence, locus of $P(h,k)$ is,
$\Rightarrow \frac{(x-1{)}^2}{1}+\frac{y^2}{4/3}=1$, which is an ellipse,
Thus length of latus rectum is $=2\frac{(\text{conjugate axis}{)}^2}{\text{transverse axis}}=2\frac{1}{2/\sqrt{3}}=\sqrt{3}$