If tangents are drawn from any point P on the | vedclass

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(A) Let the first circle be $S_1: x^2 + y^2 + 2gx + 2fy + c = 0$. Its center is $C(-g, -f)$ and its radius is $r_1 = \sqrt{g^2 + f^2 - c}$.The second

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$2\alpha$

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(A) Let the first circle be $S_1: x^2 + y^2 + 2gx + 2fy + c = 0$. Its center is $C(-g, -f)$ and its radius is $r_1 = \sqrt{g^2 + f^2 - c}$.The second circle is $S_2: x^2 + y^2 + 2gx + 2fy + c \sin^2 \alpha + (g^2 + f^2) \cos^2 \alpha = 0$. Its center is also $C(-g, -f)$.The radius of the second circle $r_2$ is given by $r_2^2 = g^2 + f^2 - (c \sin^2 \alpha + (g^2 + f^2) \cos^2 \alpha)$.$r_2^2 = g^2(1 - \cos^2 \alpha) + f^2(1 - \cos^2 \alpha) - c \sin^2 \alpha = (g^2 + f^2 - c) \sin^2 \alpha$.Thus,$r_2 = r_1 \sin \alpha$.Let $P$ be a point on $S_1$ and $T$ be the point of contact of the tangent from $P$ to $S_2$. In the right-angled triangle $	riangle PTC$,$\sin(\angle PTC) = \frac{r_2}{r_1} = \frac{r_1 \sin \alpha}{r_1} = \sin \alpha$.Therefore,$\angle PTC = \alpha$.The angle between the tangents from $P$ to $S_2$ is $2 	imes \angle PTC = 2 \alpha$.

If tangents are drawn from any point $P$ on the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ to the circle $x^2 + y^2 + 2gx + 2fy + c \sin^2 \alpha + (g^2 + f^2) \cos^2 \alpha = 0$,then the angle between the tangents is:

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