Apollonius' recipes for constructing tangent lines to parabolas, ellipses, and hyperbolas are as follows: [b]Proposition I-33[/b] Let[math] P[/math] be a point on the parabola with vertex [math]E[/math], with [math]PD[/math] perpendicular to the axis of symmetry. If [math]A[/math] is on the axis of symmetry so that [math]AE=ED[/math], then $AP$ will be tangent to the parabola at [math]P[/math]. [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].}

Use Apollonius' method to construct the tangent lines of the given curves by first determining the x intercept (for ellipses and hyperbolas) or the y intercept (for parabolas) of the tangent line according to Apollonius' propositions.