Let an ellipse with centre $(1, 0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x – axis. If its minor axis subtends angle 60° at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ______

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answer is 9.

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Detailed Solution

For calculations lets consider center at origin.
$∴$ Equation of ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

Latusrectum = 1/2
$\begin{array}{l} ∵ \frac{2 b^{2}}{a} = \frac{1}{2} \\ => 4 b^{2} = a \end{array}$
From the figure
$\begin{array}{l} b / ae = \tan 30^{\circ} \\ \frac{b}{ae} = \frac{1}{\sqrt{3}} \end{array}$
Question Image
$\begin{array}{l} ∵ 3 b^{2} = a^{2} e^{2} = a^{2} - b^{2} \\ 4 b^{2} = a^{2} \end{array}$
Solving (1) and (2), we get
$\begin{array}{l} a = 1 and b^{2} = \frac{1}{4} \\ So, ((2 a) + (2 b))^{2} = 9 \end{array}$