The length of the tangent drawn from any poin | vedclass

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(A) Let the circle $C_1$ be $x^2 + y^2 + 2gx + 2fy + \alpha = 0$ and $C_2$ be $x^2 + y^2 + 2gx + 2fy + \beta = 0$.Let $P(x_1, y_1)$ be any point on th

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$\sqrt{\beta - \alpha}$

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(A) Let the circle $C_1$ be $x^2 + y^2 + 2gx + 2fy + \alpha = 0$ and $C_2$ be $x^2 + y^2 + 2gx + 2fy + \beta = 0$.Let $P(x_1, y_1)$ be any point on the circle $C_1$.Since $P$ lies on $C_1$,we have $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + \alpha = 0$,which implies $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 = -\alpha$.The length of the tangent from a point $(x_1, y_1)$ to the circle $x^2 + y^2 + 2gx + 2fy + \beta = 0$ is given by $\sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + \beta}$.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 $\sqrt{-\alpha + \beta}$ or $\sqrt{\beta - \alpha}$.

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

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