Question

An ellipse having foci at $(3,-3)$ and $(-4,-4)$ and passing through the origin. Then the length of latus rectum is

• A. $\frac{6\sqrt{2}}{7}$
• B. $\frac{12\sqrt{2}}{7}$
• C. $\frac{16\sqrt{2}}{7}$
• D. $\frac{24\sqrt{2}}{7}$

Answer 1

The correct option is D $\frac{24\sqrt{2}}{7}$

Let $S$ and $S^{\prime }$ be the foci of the ellipse with $P$ as origin.
$SP+S^{\prime }P=2a\Rightarrow 3\sqrt{2}+4\sqrt{2}=2a\Rightarrow a=\frac{7}{\sqrt{2}}\cdots \left(1\right)S{S}^{\prime }=2ae=5\sqrt{2}\Rightarrow e=\frac{5}{7}$
Now, $e^2=1-\frac{b^2}{a^2}$
$b^2=a^2-a^2{e}^2{b}^2=\frac{49}{2}-\frac{25}{2}=12$
Length of latus rectum $=\frac{2b^2}{a}=\frac{2\times 12\times \sqrt{2}}{7}=\frac{24\sqrt{2}}{7}$