If the length of the latus rectum of an ellipse with its major axis along x-axis and center 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?

Let the equation of the ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

$∴ \frac{2 b^{2}}{a} = 8 \Rightarrow b^{2} = 4 a$ and $2 a e = 2 b \Rightarrow e = \frac{b}{a}$

Since, $e^{2} = 1 - \frac{b^{2}}{a^{2}} \Rightarrow e^{2} = 1 - e^{2} \Rightarrow e = \frac{1}{\sqrt{2}}$

Now, $\frac{1}{2} = 1 - \frac{4 a}{a^{2}} \Rightarrow a = 8$ and $b = a e = 8 \cdot \frac{1}{\sqrt{2}} = 4 \sqrt{2}$

Equation of ellipse is $\frac{x^{2}}{64} + \frac{y^{2}}{32} = 1$

The point only in option (d) i.e., $(4 \sqrt{3}, 2 \sqrt{2})$ lies on the ellipse.

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