The tangent at (3 , -2) of the circles ${x^2} + {y^2} = 13$ of the touches the circle ${x^2} + {y^2} + 2x$. and find its point of contact.

The equation x^2 + y^2 + 2x = 0 represents a circle centered at (-1, 0) with a radius of 1. To find the equation of the tangent to the first circle at (3, -2), we use the formula xx1 + yy1 = r^2, which yields 3x - 2y = 13. Solving the system of equations where 3x - 2y = 13 is tangent to the second circle results in exactly one point of contact, making the statement true.

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