Let the equation $\mathrm{ax}^2+2 \mathrm{hxy}+\mathrm{by}^2+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0$…
Let the equation $\mathrm{ax}^2+2 \mathrm{hxy}+\mathrm{by}^2+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0$ represent a point circle other than the origin. Then which one of the following conditions must hold?
$b \mathrm{c}>0$
$\mathrm{b}>0$ and $\mathrm{c} < 0$
b $ < 0$ and c $>0$
b $\leq 0$ and c $ < 0$
Solution
Given: $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 \quad \ldots$ (i)
If equation (i) represents a circle, then $a=b \& h=0$
Then $\mathrm{eq}^{\mathrm{n}}$ (i) reduces to: $b x^2+b y^2+2 g x+2 f y+c=0$
$\Rightarrow x^2+y^2+2 \cdot \frac{g}{b} x+2\left(\frac{f}{b}\right) y+\frac{c}{b}=0$ ...(ii)
$\because$ Equation (ii), represents point circle. Hence radius $r=0$
$\begin{aligned} & \Rightarrow \sqrt{\left(\frac{g}{b}\right)^2+\left(\frac{f}{b}\right)^2-\frac{c}{b}}=0 \\ & \Rightarrow \frac{c}{b}=\frac{g^2}{b^2}+\frac{f^2}{b^2} \\ & \mathrm{cb}=\mathrm{g}^2+\mathrm{f}^2\end{aligned}$
$\Rightarrow \mathrm{cb}>0\left\{\because g^2>0 \& f^2>0\right\}$