91Ó°ÊÓ

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the polar equation $$r=\frac{r_{p e r}(1+e)}{1-e \cos \theta}$$ where \(e\) is the eccentricity of the conic and \(r_{\mathrm{per}}\) is the perihelion distance measured in AU. (a) For each comet, determine whether its trajectory is elliptical, parabolic, or hyperbolic. (b) The orbit of Saturn has \(r_{\text {per }}=9.006\) and \(e=0.056\) Graph both the motion of the comet and the orbit of Saturn in the specified viewing rectangle. Encke's Comet \(\quad r_{\text {per }}=0.3317, \quad e=0.8499\) $$[-18,18,3] \text { by }[-12,12,3]$$

(a) Find three parametrizations that give the same graph as the given equation. (b) Find three parametrizations that give only a portion of the graph of the given equation. $$y=\ln x$$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+4 x+4 y^{2}-24 y=-36$$

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r=4 \sec \theta$$

Exer \(37-40:\) Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$\left\\{\begin{array}{l} x^{2}+4 y^{2}=36 \\ x^{2}+y^{2}=12 \end{array}\right.$$

Find an equation for the set of points in an xy-plane that are equidistant from the point \(P\) and the line \(L\) $$P(5,-2) ; \quad k y=4$$

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the polar equation $$r=\frac{r_{p e r}(1+e)}{1-e \cos \theta}$$ where \(e\) is the eccentricity of the conic and \(r_{\mathrm{per}}\) is the perihelion distance measured in AU. (a) For each comet, determine whether its trajectory is elliptical, parabolic, or hyperbolic. (b) The orbit of Saturn has \(r_{\text {per }}=9.006\) and \(e=0.056\) Graph both the motion of the comet and the orbit of Saturn in the specified viewing rectangle. Comet 1959 III \(\quad r_{\text {per }}=1.251, \quad e=1.003\) $$[-18,18,3] \text { by }[-12,12,3]$$

Exer. \(41-44:\) Find an equation for the set of points in an Xy-plane such that the sum of the distances from \(F\) and \(F\) is \(k\) $$F(3,0), \quad F(-3,0) ; \quad k=10$$

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r=-5$$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$-x^{2}=y^{2}-25$$