Question

The sum of the square of the length of all the chords intercepted by the lines $x+y=n$, $nϵW$ on the circle $x^2+y^2=4$ is

• A. $11$
• B. $38$
• C. $33$
• D. $44$

Answer 1

The correct option is B $38$

$x^2+y^2=4$ is given circle.

For intercept made by chord $x+y=n,$
Put $y=n-x$ in$x^2+y^2=4$

$x^2+n^2+x^2-2nx=4$
$2x^2-2nx+n^2=4$

$e{q}^n$ of line $L_1\Rightarrow$ possing through

$P(\sqrt{2},\sqrt{2})$ with slope$=-1$ is

$x+y=-2\sqrt{2}$

and $e{q}^n$ of line ${}_2\Rightarrow$ possing throngh $Q(-\sqrt{2},-\sqrt{2})$ slope $=-1$ is
$x+y=-2\sqrt{2}$

So, chords with $x+y=n,n\leftarrow W$
$\Rightarrow -2\sqrt{2}<2\sqrt{2}$
so, $n=0,1,2$
From fig 2,so, intetsept made by $x+y=2$ is $2\sqrt{4-C{E}^2}=2\sqrt{2}$
and intercept made by $x+y=1$ is $2\sqrt{4-\left(\frac{1}{2}\right)}=\sqrt{14}$

and intercept made by $x+y=0$ is $4$

$\Rightarrow (2\sqrt{2}{)}^2+(\sqrt{14}{)}^2+4^2=8+14+16=38$