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10-2. Parabola, Ellipse, Hyperbola
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Let $x^2=4 k y, k>0$ be a parabola with vertex $A$. Let $B C$ be its latusrectum. An ellipse with centre on $B C$ touches the parabola at $A$, and cuts $B C$ at points $D$ and $E$ such that $B D=D E=E C(B, D, E, C$ in that order). The eccentricity of the ellipse is
A
$\frac{1}{\sqrt{2}}$
B
$\frac{1}{\sqrt{3}}$
C
$\frac{\sqrt{5}}{3}$
D
$\frac{\sqrt{3}}{2}$
(KVPY-2018)
Solution
(c)
Given, $x^2=4 k y$
$B C$ is latusrectum.
$B C=4 k$
$B D=D E=E C$
$D E=\frac{B C}{3}=\frac{4 k}{3}$
$P$ is centre of ellipse.
$P E=\frac{2 k}{3}$
$O P^2=k$
$\because$ Eccentricity of ellipse
$\sqrt{1-\frac{P E^2}{O P^2}}=\sqrt{1-\frac{4 k^2}{9 k^2}}$
$e=\sqrt{\frac{9-4}{9}}=\frac{\sqrt{5}}{3}$
Standard 11
Mathematics
