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(A) Let the two circles have radii $r_1$ and $r_2$. Since each circle passes through the centre of the other,the distance between their centres $c_1$

Vedclass · Accepted Answer

$\frac{2 \pi}{3}$

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(A) Let the two circles have radii $r_1$ and $r_2$. Since each circle passes through the centre of the other,the distance between their centres $c_1$ and $c_2$ is equal to their radii,i.e.,$c_1 c_2 = r_1 = r_2 = d$.Consider the triangle formed by the two centres $c_1, c_2$ and one of the intersection points $P$. The sides of this triangle are $c_1 P = r_1$,$c_2 P = r_2$,and $c_1 c_2 = d$.Since $r_1 = r_2 = d$,the triangle is equilateral with all sides equal to $d$.Thus,the angle $\angle P c_1 c_2 = 60^\circ = \frac{\pi}{3}$ and $\angle P c_2 c_1 = 60^\circ = \frac{\pi}{3}$.The angle between the tangents at the point of intersection $P$ is the angle between the radii $c_1 P$ and $c_2 P$ relative to the line of centres,or more directly,the angle $	heta$ at the intersection point $P$ in the triangle $c_1 P c_2$ is $60^\circ = \frac{\pi}{3}$.However,the angle between two circles is defined as the angle between their tangents at the point of intersection. In this configuration,the angle between the tangents is $120^\circ = \frac{2 \pi}{3}$.

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

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