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 : ellipse [br] [math]e=1[/math] : parabola [br] [math]1 : 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]
A conic section is the locus of points whose distance from a fixed point (the [i]focus[/i]) and a fixed line (the [i]directrix[/i]) has a constant ratio.Consider the polar equation [math]r=\frac{\sqrt{6}}{ 2+\sqrt{6}\cos\theta}[/math] . [br]Answer the following questions.[br][b](1)[/b] Determine the type of conic represented by this curve and find its eccentricity e.[br]Assume that the focus is located at the origin.[br][b](2)[/b] Find the equation of the directrix of this conic.[br][b](3)[/b] Express this curve in Cartesian coordinates (x,y).
(1) hyperbola, [math]e=\frac{\sqrt{6}}{2}[/math][br](2) [math]x=1[/math][br](3) [math]\frac{(x-3)^2}{6}-\frac{y^2}3=1[/math]