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If the angle between the circles $x^2+y^2-4 x-6 y+k=0$ and $x^2+y^2+8 x-4 y+11=0$ is $\frac{\pi}{2}$, then the value of $k$ is
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The correct answer is:
$15$
Circles are orthogonal
$\begin{aligned} & \Rightarrow 2 g_1 g_2+2 f_1 f_2=\mathrm{C}_1+\mathrm{C}_2 \\ & \Rightarrow 2(-2)(4)+2(-3)(--2)=k+11 \\ & \Rightarrow-16+12=k+11 \Rightarrow k=-15\end{aligned}$
$\begin{aligned} & \Rightarrow 2 g_1 g_2+2 f_1 f_2=\mathrm{C}_1+\mathrm{C}_2 \\ & \Rightarrow 2(-2)(4)+2(-3)(--2)=k+11 \\ & \Rightarrow-16+12=k+11 \Rightarrow k=-15\end{aligned}$
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