Question

If the limiting points of the system of circles $x^2+y^2+2gx+\lambda (x^2+y^2+2fy+k)=0$ where $\lambda$. is a parameter. subtend a right angle at the origin, then $k/f^2=$

• A. -1
• B. 1
• C. 2
• D. None of these

Answer 1

The correct option is C 2

Limitings points are $A(\frac{-g}{1+{\lambda }_1},\frac{-{\lambda }_{1}f}{1+{\lambda }_1})$ and $B(\frac{-g}{1+{\lambda }_2},\frac{-{\lambda }_{2}f}{1+{\lambda }_2})$
where ${\lambda }_1,{\lambda }_2$ are the roots of ${\lambda }^2(f^2-k)-k\lambda +g^2=0$
OA is perpendicular to OB
$\Rightarrow \frac{{\lambda }_{1}f}{g}\times \frac{{\lambda }_{2}f}{g}=-1\Rightarrow {\lambda }_1{\lambda }_2=\frac{{-g}^2}{f^2}$
$\Rightarrow \frac{g^2}{f^2-k}=-\frac{g^2}{f^2}\Rightarrow \frac{k}{f^2}=2$