The length of the tangent drawn from any poin | vedclass

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(B) Let $P(x_1, y_1)$ be any point on the first circle $x^2 + y^2 + 2gx + 2fy + c_1 = 0$. Since $P$ lies on the circle,we have $x_1^2 + y_1^2 + 2gx_1

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$\sqrt{c - c_1}$

Vedclass · Answer

(B) Let $P(x_1, y_1)$ be any point on the first circle $x^2 + y^2 + 2gx + 2fy + c_1 = 0$. Since $P$ lies on the circle,we have $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c_1 = 0$,which implies $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 = -c_1$. The length of the tangent from a point $(x_1, y_1)$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is given by $\sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$. Substituting the value of $x_1^2 + y_1^2 + 2gx_1 + 2fy_1$ from the first equation,we get the length of the tangent as $\sqrt{-c_1 + c} = \sqrt{c - c_1}$.

The length of the tangent drawn from any point on the circle $x^2 + y^2 + 2gx + 2fy + c_1 = 0$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is:

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