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10-2. Parabola, Ellipse, Hyperbola
hard
The centre of an ellipse is $C$ and $PN$ is any ordinate and $A$, $A’$ are the end points of major axis, then the value of $\frac{{P{N^2}}}{{AN\;.\;A'N}}$ is
A
$\frac{{{b^2}}}{{{a^2}}}$
B
$\frac{{{a^2}}}{{{b^2}}}$
C
${a^2} + {b^2}$
D
$1$
Solution
(a) Let ellipse be $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$
$P = (a\cos \theta ,\,b\sin \theta ),\,A\,{\rm{ and}}\,A' \equiv ( \pm a,0),\,\,$
$N \equiv (a\cos \theta ,0),$
$PN = b\sin \theta ,$$AN = a(1 – \cos \theta ),$
$A'N = a(1 + \cos \theta )$
$\frac{{{{(PN)}^2}}}{{AN\,A'N}} = \frac{{{b^2}{{\sin }^2}\theta }}{{{a^2}(1 – \cos \theta )(1 + \cos \theta )}} = \frac{{{b^2}}}{{{a^2}}}$.
Standard 11
Mathematics
