Consider ellipses $$E_k: kx^2 + k^2y^2 = 1$$, $$k = 1, 2, \ldots, 20$$. 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 $$\sum_{k=1}^{20} \frac{1}{r_k^2}$$ is

The ellipse $$E_k: kx^2 + k^2y^2 = 1$$ can be written as:

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

So semi-major axis $$a = \frac{1}{\sqrt{k}}$$ and semi-minor axis $$b = \frac{1}{k}$$.

The four chords join the endpoints of the major axis $$(±a, 0)$$ and the minor axis $$(0, ±b)$$, forming a rhombus.

The radius of the inscribed circle of this rhombus is:

$$r_k = \frac{ab}{\sqrt{a^2 + b^2}}$$

$$r_k = \frac{\frac{1}{\sqrt{k}} \cdot \frac{1}{k}}{\sqrt{\frac{1}{k} + \frac{1}{k^2}}} = \frac{\frac{1}{k^{3/2}}}{\sqrt{\frac{k+1}{k^2}}} = \frac{\frac{1}{k^{3/2}}}{\frac{\sqrt{k+1}}{k}} = \frac{1}{\sqrt{k}\sqrt{k+1}} = \frac{1}{\sqrt{k(k+1)}}$$

Therefore:

$$\frac{1}{r_k^2} = k(k+1)$$

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

$$= \frac{20 \times 21 \times 41}{6} + \frac{20 \times 21}{2} = 2870 + 210 = 3080$$