Question

The equation of the circle for which $xy+2y+2x+4=0$ are normals and is passing through the point $(-1,1)$ is

• A. $x^2+y^2+x-y=0$
• B. $x^2+y^2+2x-2y+2=0$
• C. $x^2+y^2+4x+4y-2=0$
• D. $x^2+y^2-5x-3y=0$

Answer 1

The correct option is C $x^2+y^2+4x+4y-2=0$

Given, $xy+2y+2x+4=0$
$\Rightarrow (x+2)(y+2)=0$
$\Rightarrow x=-2$ and $y=-2$ are two normals of circle.
As we know, all normals of circle passes through centre of circle.
$\therefore$ centre $c\equiv (-2,-2)$
$\therefore$ centre of circel is intersection of $x=-2$ and $y=-2$ lines.
$\therefore$ radius of circle is distance between $(-2,-2)$ and $(-1,1)$
$\therefore$ radius $=\sqrt{10}$
$\therefore$ equation of circle is
$(x+2{)}^2+(y+2{)}^2=(\sqrt{10}{)}^2$
$\Rightarrow x^2+y^2+4x+4y-2=0.$