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(D) The circle is $x^2 + y^2 = 16$,so its radius $r = 4$. The distance $p$ from the center $(0, 0)$ to the line $x + y - n = 0$ is $p = \frac{|0 + 0 -

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$210$

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(D) The circle is $x^2 + y^2 = 16$,so its radius $r = 4$. The distance $p$ from the center $(0, 0)$ to the line $x + y - n = 0$ is $p = \frac{|0 + 0 - n|}{\sqrt{1^2 + 1^2}} = \frac{n}{\sqrt{2}}$.For a chord to exist,$p The length of the chord $L$ is given by $L = 2\sqrt{r^2 - p^2} = 2\sqrt{16 - \frac{n^2}{2}} = \sqrt{64 - 2n^2}$.The square of the length is $L^2 = 64 - 2n^2$.For $n = 1, L^2 = 64 - 2(1) = 62$.For $n = 2, L^2 = 64 - 2(4) = 56$.For $n = 3, L^2 = 64 - 2(9) = 46$.For $n = 4, L^2 = 64 - 2(16) = 32$.For $n = 5, L^2 = 64 - 2(25) = 14$.The sum of the squares is $62 + 56 + 46 + 32 + 14 = 210$.

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

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