Consider ellipse Ek: x^2k + y^2k = 1, for k = 1, 2,, 20. Let Ck be the circle which touches the four chords joining the

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Consider ellipse Ek: x^2k + y^2k = 1, for k = 1, 2,, 20. Let Ck be the circle which touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse…

Accepted Answer

Given: the solution states that the correct option is C and concludes that $\sum_{k=1}^{20} r_k^2 = 3080$ .

Find: The value of $\sum_{k=1}^{20} r_k^2$ .

From the extracted solution text:

r_k = \frac{1}{\sqrt{K + K^2}}

Hence,

r_k^2 = \frac{1}{K + K^2}

The solution then writes:

\sum_{k=1}^{20} r_k^2 = \sum_{k=1}^{20} \left( \frac{1}{K + K^2} \right) = 210 + 10 \times 70 + 10 \times 70 = 3080

Therefore, according to the solution, the value of $\sum_{k=1}^{20} r_k^2$ is $3080$ . Hence, the correct option is C.

Note: The intermediate derivation shown in the solution is internally inconsistent, but its stated final conclusion clearly selects option C.

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