The condition that the pair of tangents drawn from the origin to the circle $x^2+y^2+2gx+2fy+c=0$. may be at right angles is

Equation to the pair of tangents drawn from $(0,0)\text{ to }{x}^2+y^2+2gx+2fy+c=0$ is

$(gx+fv+c{)}^2=c\left(x^2+y^2+2gx+2fy+c\right)$

$\Rightarrow g^2{x}^2+2fgxy+f^2{y}^2+2cgx+2cfy+c^2=c{x}^2+c{y}^2+2cgx+2fy+c^2$

$\Rightarrow \left(g^2-c\right)x^2+2fgxy+\left(f^2-c\right)y^2=0$

The tangents are at right angles $\Rightarrow \text{ coeff. of }{x}^2+\text{ coeff. of }{y}^2=0\Rightarrow g^2-c+f^2-c=0$

$\Rightarrow g^2+f^2=2c$