The sum of the squares of the lengths of the chords intercepted on the circle, $x^2+y^2=16,$ by the lines, $x+y=n, n\in N,$ where $N$ is the set of all natural numbers is:

The length of the perpendicular from centre $\left(0,0\right)$  to the line $x+y=n$ is $\left|\frac{0+0-n}{\sqrt{1^2+I^2}}\right|=\frac{n}{\sqrt{2}}$

Length of intercept $=2\sqrt{4^2-{\left(\frac{n}{\sqrt{2}}\right)}^2}=\sqrt{64-2n^2}$

$\Rightarrow$ Possible values of $n$ are $1,2,\dots .5$

$\Rightarrow$ Sum of the square of lengths of intercepts

$=\underset{n=1}{\overset{5}{\Sigma }}\left(64-2n^2\right)=64\times 5-2\underset{n=1}{\overset{5}{\Sigma }}{n}^2$

$=320-2\underset{n=1}{\overset{5}{\Sigma }}{n}^2=320-110=210$