Question

The equation of the circle which passes through origin and cuts off chords of length 2 on the lines $x=y$ and $x=-y$ is

• A. $x^2+y^2\pm 2\sqrt{2}y=0,x^2+y^2\pm 2\sqrt{2}x=0$
• B. $x^2+y^2\pm 3\sqrt{3}x=0,x^2+y^2$ $\pm 3\sqrt{3}y=0$
• C. $x^2+y^2\pm 4\sqrt{2}x=0,x^2+y^2\pm 4\sqrt{2}y=0$
• D. $x^2+y^2\pm 4\sqrt{2}x=0,x^2+y^2\pm 4\sqrt{3}y=0$

Answer 1

The correct option is A $x^2+y^2\pm 2\sqrt{2}y=0,x^2+y^2\pm 2\sqrt{2}x=0$

Let $x^2+y^2+2gx+2fg=0$ is a circle passing thorugh $(0,0)$

and this circle cut off chord of length $2$ on

the line $x=yandx=-y$

So,$x=y,$ $2x^2+2x(g+f)=0$

$x=0,$ $x=-(g+f)$

$\angle 1=0,y=-(g+f)$

$2=\sqrt{2(g+f{)}^2}$

$g+f=\sqrt{2}$ ---(1) or $g+f=-\sqrt{2}$ ---(4)

for $x=-y,$ $2x^2+2x(g-f)=0$

$x=0$ or $x=f-g$

and $y=0$ $y=g-f$

then $2=\sqrt{2(g-f{)}^2}$

$g-f=\sqrt{2}$ ---(2) or $g-f=-\sqrt{2}$ ----(3)

from (1) & (2),$g=\sqrt{2},f=0$

from (1) & (3),$g=0,f=\sqrt{2}$

from (4) & (2),$g=0,f=-\sqrt{2}$

from (4) & (3),$g=-\sqrt{2},f=0$

So, $e{q}^n$ of circles are

$x^2+y^2\pm 2\sqrt{2}y=0$ and $x^2+y^2\pm 2\sqrt{2}x=0$