Let an ellipse with centre (1, 0)and latus rectum of length 1/2 have its major axis along x-axis. If its minor axis subtends an angle 60^o at the foci then the square of the sum of the lengths of its minor axis and major axes is equal to

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answer is 9.

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Detailed Solution

$\begin{array}{l} L a t u s r e c t r u m = \frac{2 b^{2}}{a} = \frac{1}{2}, \\ \Rightarrow b^{2} = \frac{a}{4} \tan 30^{\circ} = \frac{b}{a e} \Rightarrow \frac{1}{3} = \frac{b^{2}}{a^{2} - b^{2}} \Rightarrow a^{2} - b^{2} = 3 b^{2} \Rightarrow b^{2} = \frac{a^{2}}{4} \\ \Rightarrow a = 1, b^{2} = \frac{1}{4} \Rightarrow b = \frac{1}{2} \\ \Rightarrow (2 a + 2 b)^{2} = 9 \end{array}$