The angle between the two circles, each passing through the centre of the other is

1. $\frac{2 \pi}{3}$
2. $\frac{\pi}{2}$
3. $\frac{\pi}{6}$
4. $\pi$

Solution

Let $\theta$ be the angle between two circles. As each circle passing through the centre at each other. Then, $c_1 c_2=r_1=r_2$
From the figure, $ \begin{aligned} & \cos \theta=\frac{r_1^2+r_2^2-c_1 c_2}{2 r_1 r_2} \\ & \cos \theta=\frac{r_1^2+r_2^2-r_1^2}{2 r_1 r_2} \\ & \cos \theta=\frac{1}{2} \\ & \theta=\frac{\pi}{3} \end{aligned} $ Hence, angle between two circle is either $\frac{\pi}{3}$ or $ \pi-\frac{\pi}{3}=\frac{2 \pi}{3} $

Asked in: AP EAMCET 2017 (26 Apr Shift 1)

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