Consider ellipses E k : k x 2 + k 2 y 2 = 1 , k = 1 , 2 , … , 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

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Solution

Q.
Consider ellipses $E_{k} : k x^{2} + k^{2} y^{2} = 1, k = 1, 2, \dots, 20 .$ 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 $\sum_{k = 1}^{20} \frac{1}{r_{k}^{2}}$ is [2023]

1 3080

2 3210

3 3320

4 2870

Ans.
(1)

$k x^{2} + k^{2} y^{2} = 1$

$\frac{x^{2}}{1 / k} + \frac{y^{2}}{1 / k^{2}} = 1$

Now, equation of line $A B$ is

$\frac{x}{1 / \sqrt{k}} + \frac{y}{1 / k} = 1 \Rightarrow \sqrt{k} x + k y = 1$

$r_{k} = ⊥ r$ distance of (0, 0) from line AB

$r_{k} =$ $| \frac{(0 + 0 - 1)}{\sqrt{k + k^{2}}} |$ $= \frac{1}{\sqrt{k + k^{2}}}$

$\frac{1}{r_{k}^{2}} = k + k^{2}$ $\Rightarrow \sum_{k = 1}^{20} \frac{1}{r_{k}^{2}} = \sum_{k = 1}^{20} (k + k^{2}) = \sum_{k = 1}^{20} k + \sum_{k = 1}^{20} k^{2}$

$= \frac{20 \cdot 21}{2}$ $+ \frac{20 \cdot 21 \cdot 41}{6} = 210 + 10 \times 7 \times 41$

$= 210 + 2870 = 3080$

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