Descartes' Method of Tangents

The circle through [math]A[/math] with center [math]O[/math] intersects the curve [math]y=f(x)[/math] at two points [math]A[/math] and [math]B[/math]. Drag [math]O[/math], the center of the circle, until [math]A[/math] and [math]B[/math] coincide. The resulting circle will then be tangent to the curve.

The tangent to the curve [math]y=f(x)[/math] is the same as the tangent to the circle, which is perpendicular to the radius. Use this fact to find an equation to the tangent to [math]y=f(x)[/math] at [math]A=(1,2)[/math]. Then again at [math]A=(9,6)[/math].