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

Let any point on the first circle be (x, y), which when substituted into its equation gives x squared plus y squared plus 2gx plus 2fy equals negative p. The length of the tangent from an external point to a second circle is found by substituting the point into the second circle's equation and taking the square root, meaning the length squared equals x squared plus y squared plus 2gx plus 2fy plus q. Substituting negative p into this expression yields a tangent length of the square root of q minus p.

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