The sum of the squares of the lengths of the chords intercepted on the circle x^2+y^2=16 b…

Question

The sum of the squares of the lengths of the chords intercepted on the circle $x^2+y^2=16$ by the lines $x+y=n$, $n\in\mathbb N$, is

Accepted Answer

Distance from origin to $x+y=n$: $\dfrac n{\sqrt2}$. Chord exists only while $\dfrac{n^2}2<16\Rightarrow n<4\sqrt2\approx5.66$, so $n=1,\dots,5$.

Chord$^2=4\left(16-\dfrac{n^2}2\right)$.

$$\sum_{n=1}^5\left[64-2n^2\right]=64(5)-2\cdot\frac{5\cdot6\cdot11}6=320-110=210.$$

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