Slope of Tangent Line to Circle at Point on Circumference

In a coordinate plane, there is a circle with center at point C(1, -2) and radius 5 units. A tangent line to this circle at a point P(x, y) on the circumference is drawn. If the coordinates of point P are determined by the equation of the circle, what will be the slope of the tangent line at point P?

Recall that the equation of a circle with center (h, k) and radius r is given by:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

For this circle, substitute the known center and radius to establish the equation:

$$ (x - 1)^2 + (y + 2)^2 = 5^2 $$

Utilize implicit differentiation to find the slope of the tangent line.