A conic section can be represented by the polar equation [math]r=\frac{ea}{1+e \cos \theta}[/math] .[br]Use the sliders for e and a in the applet below, then answer the questions that follow.[br]
How is the curve classified as a conic section depending on the value of e?
[math]0<e<1[/math] : ellipse [br] [math]e=1[/math] : parabola [br] [math]1<e[/math] : hyperbola
Which geometric property of the curve is determined by the parameter a?
The parameter a determines the [i]size[/i] (scale) of the curve.
Express the distance [math]FP[/math] from point P to the focus and the distance [math]PH[/math] from P to the directrix using the polar coordinates [math](r,\ \theta)[/math] of P and the constant a.[br]Substitute these expressions into the definition [math]e=\frac{FP}{PH}[/math] and simplify the equation to obtain an expression for r.
FP=r, PH=[math]a-r \cos \theta[/math],[br][math]r=\frac{ea}{1+e\cos \theta}[/math]