Question

If the locus of the middle point 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 $\sqrt{3}$

Given ellipse is $\frac{x^2}{3}+\frac{y^2}{4}=1$
Let the middle point of chord be $(h,k)$
Then equation of chord is $T=S_1$
$\Rightarrow \frac{xh}{3}+\frac{yk}{4}=\frac{h^2}{3}+\frac{k^2}{4}$
As chord passes through $(2,0),$
$\frac{2h}{3}=\frac{h^2}{3}+\frac{k^2}{4}$
$\Rightarrow \frac{(h-1{)}^2}{1}+\frac{k^2}{4/3}=1$
Hence, equation of the ellipse $A$ is
$\frac{(x-1{)}^2}{1}+\frac{y^2}{4/3}=1$
Here, $a=1,b=\frac{2}{\sqrt{3}},b>a$
Length of latus rectum $=\frac{2a^2}{b}=\frac{2\times 1}{2/\sqrt{3}}=\sqrt{3}$