Question

Let the length of the latus rectum of an ellipse with its major-axis along $x$-axis and centre at the origin, be $8$. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

• A. $(4\sqrt{2},2\sqrt{2})$
• B. $(4\sqrt{2},2\sqrt{3})$
• C. $(4\sqrt{3},2\sqrt{3})$
• D. $(4\sqrt{3},2\sqrt{2})$

Answer 1

The correct option is D $(4\sqrt{3},2\sqrt{2})$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$
Length of latus ractum $=\frac{2b^2}{a}=8\cdots \left(1\right)$
and distance between foci $=$ length of minor axis
$2ae=2b$
$\Rightarrow ae=b\cdots \left(2\right)$
We know that,
$a^2{e}^2=a^2-b^2$
From equation $\left(1\right)$, we get
$\Rightarrow b^2=a^2-b^2\Rightarrow 2b^2=a^2\cdots \left(3\right)$

From equation $\left(1\right)$ and $\left(3\right)$, we have
$\frac{a^2}{a}=8\Rightarrow a=8$
From equation $\left(3\right)$, we get
$b^2=32$

Thus, equation of ellipse is
$\Rightarrow \frac{x^2}{64}+\frac{y^2}{32}=1$
Clearly, $(4\sqrt{3},2\sqrt{2})$ lies on the ellipse.