Let the length of the latus rectum of an ellipse with its major axis along $x$ -axis and centre at the…
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?
$(4 \sqrt{2}, 2 \sqrt{2})$
$(4 \sqrt{3}, 2 \sqrt{2})$
$(4 \sqrt{3}, 2 \sqrt{3})$
$(4 \sqrt{2}, 2 \sqrt{3})$
Solution
Let the ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
Then, $\frac{2 b^{2}}{a}=8,2 a e-b^{2}$ and $b^{2}=a^{3}\left(1-e^{2}\right)$
$\Rightarrow a=8, b^{2}=32$
Then, the equation of the ellipse
$\frac{x^{2}}{64}+\frac{y^{2}}{32}=1$
Hence, the point $(4 \sqrt{3}, 2 \sqrt{2})$ lies on the ellipse.