Apollonius' recipe for construction tangent lines to ellipses and hyperbolas is as follows: [b]Proposition I-34[/b] Let [math]P[/math] be a point on an ellipse or hyperbola, [math]PB[/math] the perpendicular from the point to the main axis. Let [math]G[/math] and [math]H[/math] be the intersections of the axis with the curve and choose [math]A[/math] on the axis so that [math]\frac{AH}{AG}=\frac{BH}{BG}[/math]. Then [math]AP[/math] will be tangent to the curve at [math]P[/math].

In the applet above, drag [math]H[/math] and/or [math]C[/math] to change the shape of the hyperbola. Drag [math]P[/math] to see the tangent line at any point on the hyperbola.