Let the length of the latus rectum of an ellipse with its major axis along the x-axis and …

Question

Let the length of the latus rectum of an ellipse with its major axis along the $x$-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?

Accepted Answer

$\dfrac{2b^2}a=8$ and $2ae=2b\Rightarrow ae=b$. With $b^2=a^2(1-e^2)=a^2-(ae)^2=a^2-b^2\Rightarrow2b^2=a^2$.

From $2b^2=8a\Rightarrow b^2=4a$, and $2b^2=a^2\Rightarrow a^2=8a\Rightarrow a=8,\ b^2=32$.

$$\frac{x^2}{64}+\frac{y^2}{32}=1.$$

Checking $(4\sqrt3,2\sqrt2)$: $\dfrac{48}{64}+\dfrac8{32}=0.75+0.25=1$ .

Solutions · 1