JEE MAIN - Mathematics (2023 - 15th April Morning Shift - No. 16)
Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $\mathrm{x}$-axis. If its minor axis subtends an angle $60^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ____________.
Answer
9
Explanation
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$$ \begin{aligned} & \frac{2 b^2}{a}=\frac{1}{2}, \quad \tan 30^{\circ}=\frac{b}{a e} \\\\ & b^2=\frac{a}{4}, \frac{1}{3}=\frac{b^2}{a^2-b^2} \Rightarrow a^2-b^2=3 b^2 \Rightarrow b^2=\frac{a^2}{4} \\\\ & \Rightarrow \quad a=1, b^2=\frac{1}{4} \Rightarrow b=\frac{1}{2} \\\\ & \Rightarrow \quad(2 a+2 b)^2=9 \end{aligned} $$
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