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10-2. Parabola, Ellipse, Hyperbola
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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 :
A$\frac{4}{\sqrt{17}}$
B$\frac{\sqrt{3}}{16}$
C$\frac{3}{\sqrt{19}}$
D$\frac{\sqrt{5}}{7}$
(JEE MAIN-2025)
Solution
$2 b=\frac{1}{4}(2 ae )$
$\frac{ b }{ a }=\frac{ e }{4}$
$e =\sqrt{1-\frac{ b ^2}{ a ^2}}$
$e =\sqrt{1-\frac{ e ^2}{16}}$
$e ^2\left(1+\frac{1}{16}\right)=1$
$e =\frac{4}{\sqrt{17}}$
$\frac{ b }{ a }=\frac{ e }{4}$
$e =\sqrt{1-\frac{ b ^2}{ a ^2}}$
$e =\sqrt{1-\frac{ e ^2}{16}}$
$e ^2\left(1+\frac{1}{16}\right)=1$
$e =\frac{4}{\sqrt{17}}$
Standard 11
Mathematics
