The locus of centres of the circles, which cut the circles $x^2+y^2+4 x-6 y+9$ and $x^2+y^2-5 x+4 y+2=0$ orthogonally, is

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Aicharya · Accepted Answer

Let the circle be

Cuts the circle $\begin{aligned} & x^2+y^2+4 x-6 y+9=0 \text { and } \\ & x^2+y^2-5 x+4 y+2=0 \end{aligned}$ orthogonally. For circle $\begin{aligned} & x^2+y^2+4 x-6 y+9=0 \\ & 2\left(g_1 g_2+f_1 f_2\right)=c_1+c_2 \\ & 2[(g)(-2)+(f)(3)]=c+9 \end{aligned}$

For circle, $\begin{aligned} & x^2+y^2-5 x+4 y+2=0 \\ & 2\left[(g)\left(\frac{5}{2}\right)+(f)(-2)\right]=c+2 \end{aligned}$

On subtracting Eq. (iii) from Eq. (ii), we get $\begin{aligned} & -9 g+10 f=7 \\ & 9 g-10 f=-7 \end{aligned}$ Replace $g$ and $f$ by $x$ and $y$ to get the locus of centre, $\begin{aligned} & 9 x-10 y=-7 \\ & \Rightarrow \quad 9 x-10 y+7=0 \end{aligned}$

The locus of centres of the circles, which cut the circles $x^2+y^2+4 x-6 y+9$ and $x^2+y^2-5 x+4 y+2=0$…

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