Question

Find the sum of square of the length of the chords intercepted by the line $x+y=n;n\in N$ on the circle $x^{2}+y^{2}=4$ .

Answer 1

$AB=2AM$
$AB^{2}=4AM^{2}$
$4\left(4 - \frac{n^{2}}{2}\right)=2\left(8 - n^{2}\right),n\in N$
$n=1$ or $n=2$ ( $\because $ length should be $+ve$ )
Hence required sum
$=2\left(8 - 1^{2} + 8 - 2^{2}\right)$
$=2\times 11=22$