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Consider ellipses $E _{ k }: kx ^2+ k ^2 y ^2=1, k =1,2, \ldots$,$20$. Let $C _{ k }$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$, If $r_k$ is the radius of the circle $C _{ k }$, then the value of $\sum \limits_{ k =1}^{20} \frac{1}{ I _{ k }^2}$ is $.......$.
$3080$
$3210$
$3320$
$2870$
Solution
$Kx x ^2+ K ^2 y ^2=1$
$\frac{ x ^2}{1 / K }+\frac{ y ^2}{1 / K ^2}=1$
Now
Equation of
$A _1 B _2 ; \frac{ x }{1 / \sqrt{ K }}+\frac{ y }{1 / K }=1 \Rightarrow \sqrt{ K } x + Ky =1$
$r_K=\perp r$ distance of $(0,0)$ from line $A_1 B_1$
$r _{ L }=\left|\frac{(0+0-1)}{\sqrt{ K + K ^2}}\right|=\frac{1}{\sqrt{ K + K ^2}}$
$\frac{1}{ r _{ K }^2}= K + K ^2 \Rightarrow \sum \limits_{ k =1}^{20} \frac{1}{ r _{ K }^2}=\sum \limits_{ K =1}^{20}\left( K + K ^2\right)$
$=\frac{20 \times 21}{2}+\frac{20.21 .41}{6} K +\sum \limits_{ K =1}^{20} K ^2$
$=210+10 \times 7 \times 41$
$=210+2870$
$=3080$
