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 and the equation of the tangent line is, show that:

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

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

(b) The point of tangency is (-r²m/b, r²/b).

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

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