Q32 / 75 · JEE Main 2025 (2 Apr Shift 2) · Mathematics

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

  • Correct answer
    A
    $\frac{4}{\sqrt{17}}$
  • B
    $\frac{\sqrt{3}}{16}$
  • C
    $\frac{3}{\sqrt{19}}$
  • D
    $\frac{\sqrt{5}}{7}$

Solution

$\begin{aligned} & 2 \mathrm{~b}=\frac{1}{4}(2 \mathrm{ae}) \\ & \frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{e}}{4} \\ & \mathrm{e}=\sqrt{1-\frac{\mathrm{b}^2}{\mathrm{a}^2}} \\ & \mathrm{e}=\sqrt{1-\frac{\mathrm{e}^2}{16}} \\ & \mathrm{e}^2\left(1+\frac{1}{16}\right)=1 \\ & \mathrm{e}=\frac{4}{\sqrt{17}}\end{aligned}$

Hence, the correct answer is option: 'A'.

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