The Greek Method The Greek method for finding the equation of the tangent line to a circle uses the fact that at any point on a circle the lines containing the center and the tangent line are perpendicular (see Problem). Use this method to find an equation of the tangent line to the circle x2 + y2 = 9 at the point .
Problem
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.
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