Consider ellipses E _ k : k x ^ 2 + k ^ 2 y ^ 2 = 1, k = 1, 2, , 20...

Given,

${E}_{k}: k{x}^{2} + {k}^{2}{y}^{2} = 1$

$⇒ {E}_{k}: \dfrac{{x}^{2}}{{\left(\dfrac{1}{\sqrt{k}}\right)}^{2}} + \dfrac{{y}^{2}}{{\left(\dfrac{1}{k}\right)}^{2}} = 1$

Now equation of the chord joining the points \left(\dfrac{1}{\sqrt{k}}, 0\right) \text{ and } \left(0, \dfrac{1}{k}\right)$will be,</p><p>${L}_{k}: \dfrac{x}{\left(\dfrac{1}{\sqrt{k}}\right)} + \dfrac{y}{\left(\dfrac{1}{k}\right)} = 1$</p><p>$⇒ \sqrt{k}x + ky - 1 = 0$</p><p>Now${r}_{k} =$Perpendicular distance of${L}_{k}$from$( 0, 0 )$we get,</p><p>${r}_{k} = \left|\dfrac{- 1}{\sqrt{k + {k}^{2}}}\right|$</p><p>$⇒ {r}_{k}^{2} = \dfrac{1}{k + {k}^{2}}$</p><p>Now putting the value of${{r}_{k}}^{2} in \mathop{∑}_{k = 1}^{20}1{r}_{k}^{2}$we get,</p><p>$\mathop{∑}_{k = 1}^{20}1{r}_{k}^{2} = \mathop{∑}_{k = 1}^{20}k + {k}^{2} = \dfrac{20 \times 21}{2} + \dfrac{20 \times 21 \times 41}{6}$</p><p>$= 210 + 2870= 3080