Is $2 x + 3 y = 0$ a tangent from $(- 3, 2)$ to circle $x^{2} + y^{2} + 10 x + 2 y + 13$ ? How to check like this if it is correct or not?

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Solution

$x^{2} + y^{2} + 10 x + 2 y + 13 = (x + 5)^{2} + (y + 1)^{2} - 13 = 0$
Therefore, Center is $(- 5, - 1)$ and radius is $\sqrt{13}$
The distance of the center from the line $2 x + 3 y = 0$ is $\frac{| 2 (- 5) + 3 (- 1) |}{\sqrt{2^{2} + 3^{2}}}$ is $\sqrt{13}$ .
Therefore, the line is the tangent to the circle.

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