The tangent line to a circle may be defined as the line that intersects the circle in a si...

The tangent line to a circle may be defined as the line that intersects the circle in a single point, called the point of tangency. See the figure.

If the equation of the circle is x2 + y2 = r2 and the equation of the tangent line is y = mx + b, show that:

(a) r2(1 + m2) = b2

[Hint: The quadratic equation x2 + (mx + b)2 = r2 has exactly one solution.]

(b) The point of tangency is .

(c) The tangent line is perpendicular to the line containing the center of the circle and the point of tangency.