Q. 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 :
Solution:
$p= \frac{n}{\sqrt{2}} \frac{n}{\sqrt{2}} < 4 \Rightarrow n =1, 2,3,4,5. $
Length of chord $AB =2\sqrt{16 - \frac{n^{2}}{2}} $
$ = \sqrt{64-2n^{2}} =\ell\left(say\right) $
For $n = 1, \ell^2 = 62$
$n = 2, \ell^2 = 56$
$n = 3, \ell^2 = 46$
$n = 4, \ell^2 = 32$
$n = 5, \ell^2 = 14$
$\therefore $ Required sum = $62 + 56 + 46 + 32+14 = 210$
