The sum of the squares of the lengths of the chords intercepted on...

The length of the perpendicular from centre $\left(0, 0\right)$ to the line $x + y = n$ is \left|\dfrac{0 + 0 - n}{\sqrt{{1}^{2} + {I}^{2}}}\right| = \dfrac{n}{\sqrt{2}}$</p><p>Length of intercept$= 2\sqrt{{4}^{2} - {\left(\dfrac{n}{\sqrt{2}}\right)}^{2}} = \sqrt{64 - 2{n}^{2}}$</p><p>$⇒$Possible values of$n$are$1, 2, \ldots . 5$</p><p>$⇒$Sum of the square of lengths of intercepts</p><p>$= \mathop{Σ}_{n = 1}^{5}\left(64 - 2{n}^{2}\right) = 64 \times 5 - 2\mathop{Σ}_{n = 1}^{5}{n}^{2}$</p><p>$= 320 - 2\mathop{Σ}_{n = 1}^{5}{n}^{2} = 320 - 110 = 210