QJEEM23Ma3ad9: JEE Main 2023 Mathematics - Class 11 JEEM23Ma3ad9 Solution: Let an ellipse with centre and latus rectum of length have its…

Question

Verified Solution • 2025–26 Session moderate

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 ____________.

Accepted Answer

\begin{aligned} \frac{2 b^{2}}{a} = \frac{1}{2}, \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 a = 1, b^{2} = \frac{1}{4} \Rightarrow b = \frac{1}{2} \\ \Rightarrow (2 a + 2 b)^{2} = 9 \end{aligned}

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