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If the angle between the circles $x^2+y^2-2 x+2 y+1=0$ and $x^2+y^2+2 x-2 y+k=0$ is $\frac{\pi}{3}$, then
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Verified Answer
The correct answer is:
$\mathrm{k}$ is an irrational number
Angle between circles :
$$
\begin{aligned}
& \cos \theta=\frac{r_1^2+r_2^2-d^2}{2 r_1 r_2} \\
& \cos \frac{\pi}{3}=\frac{1+2-K-8}{2 \cdot(1) \sqrt{2-K}}=\frac{1}{2} \\
& \Rightarrow \sqrt{2-K}=-5-K \\
& \Rightarrow 2-K=25+K^2+10 K \\
& \Rightarrow K^2+11 K+23=0 \Rightarrow K=\frac{-11 \pm \sqrt{29}}{2}
\end{aligned}
$$
$\therefore \quad K$ is an irrational number.
$$
\begin{aligned}
& \cos \theta=\frac{r_1^2+r_2^2-d^2}{2 r_1 r_2} \\
& \cos \frac{\pi}{3}=\frac{1+2-K-8}{2 \cdot(1) \sqrt{2-K}}=\frac{1}{2} \\
& \Rightarrow \sqrt{2-K}=-5-K \\
& \Rightarrow 2-K=25+K^2+10 K \\
& \Rightarrow K^2+11 K+23=0 \Rightarrow K=\frac{-11 \pm \sqrt{29}}{2}
\end{aligned}
$$
$\therefore \quad K$ is an irrational number.
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