Find the lengths of the tangents drawn to the circle $3 x^{2} + 3 y^{2} - 7 x - 6 y = 12$ from the point (6, 7) .

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Solution

For a circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , length of tangent from a external point $(x_{1}, y_{1})$ is given by
$L = \sqrt{{x_{1}}^{2} + {y_{1}}^{2} + 2 g x_{1} + 2 f y_{1} + c}$
For the given circle $3 x^{2} + 3 y^{2} - 7 x - 6 y = 12$ ,
$L = \sqrt{36 + 49 + 2 (\frac{- 7}{6}) (6) + 2 (- 1) (7) - 4}$
$L = \sqrt{63}$

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