Chapter 10

(a) Describe the graph of a curve \(C\) that has the parametrization $$x=-2+3 \sin t, \quad y=3-3 \cos t ; \quad 0 \leq t \leq 2 \pi$$ (b) Change the parametrization to $$x=-2-3 \sin t, \quad y=3+3 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a). (c) Change the parametrization to $$x=-2+3 \sin t, \quad y=3+3 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a).

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 34,0), \quad\) asymptotes \(y=\pm \frac{3}{5} x\)

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$(x+2)^{2}+y^{2}=4$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{2}{3}, \quad\) vertices on the \(y\) -axis, passing through \((1,4)\)

Find an equation of the parabola that satisfies the given conditions. $$\text { Vertex } V(4,7), \quad \text { focus } F(4,2)$$

Find a polar equation of the parabola with focus at the pole and the given vertex. $$V\left(4, \frac{\pi}{2}\right)$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm 5, \quad\) asymptotes \(y=\pm 2 x\)

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}+(y+3)^{2}=9$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm 2\) \(y\) -intercepts \(\pm \frac{1}{3}\)

Find an equation of the parabola that satisfies the given conditions. Vertex at the origin, symmetric with respect to the \(y\) -axis, and passing through the point \((2,-3)\)

Find a polar equation of the parabola with focus at the pole and the given vertex. $$V(5,0)$$

Curves \(C_{1}, C_{2}, C_{3},\) and \(C_{4}\) are given parametrically, for \(t\) in \(\mathbb{R}\). Sketch their graphs, and indicate orientations. $$\begin{aligned}&C_{1}: x=t, \quad y=1-t\\\&C_{2}: x=1-t^{2}, \quad y=t^{2}\\\&C_{3}: x=\cos ^{2} t, \quad y=\sin ^{2} t\\\&C_{4}: x=\ln t-t, \quad y=1+t-\ln t, t>0\end{aligned}$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. \(y\) -intercepts \(\pm 2\) asymptotes \(y=\pm \frac{1}{4} x\)

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}+(y-1)^{2}=1$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm \frac{1}{2}, \quad y\) -intercepts \(\pm 4\)

Find an equation of the parabola that satisfies the given conditions. Vertex at the origin, symmetric with respect to the \(y\) -axis, and passing through the point \((6,3)\)

The parametric equations specify the position of a moving point \(P(x, y)\) at time \(t\). Sketch the graph, and indicate the motion of \(P\) as \(t\) increases. (a) \(x=\cos t, \quad y=\sin t, \quad 0 \leq t \leq \pi\) (b) \(x=\sin t, \quad y=\cos t, \quad 0 \leq t \leq \pi\) (c) \(x=t\) \(y=\sqrt{1-t^{2}} ; \quad-1 \leq t \leq 1\)

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertical transverse axis of length \(10,\) conjugate axis of length 14

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$(x+2)^{2}+(y-3)^{2}=13$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Horizontal major axis of length \(8,\) minor axis of length 5

The parametric equations specify the position of a moving point \(P(x, y)\) at time \(t\). Sketch the graph, and indicate the motion of \(P\) as \(t\) increases. (a) \(x=t^{2}, \quad y=1-t^{2} ; \quad 0 \leq t \leq 1\) (b) \(x=1-\ln t, \quad y=\ln t, \quad 1 \leq t \leq e\) (c) \(x=\cos ^{2} t, \quad y=\sin ^{2} t, \quad 0 \leq t \leq 2 \pi\)

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Horizontal transverse axis of length \(6,\) conjugate axis of length 2

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$(x-3)^{2}+(y+4)^{2}=25$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertical major axis of length \(7,\) minor axis of length 6

Find an equation of the parabola that satisfies the given conditions. Vertex \(V(3,-2),\) axis parallel to the \(x\) -axis, and \(y\) -intercept 1

Kepler's first law asserts that planets travel in elliptical orbits with the sun at one focus. To find an equation of an orbit, place the pole \(O\) at the center of the sun and the polar axis along the major axis of the ellipse (see the figure). (a) Show that an equation of the orbit is $$r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}$$ where \(e\) is the eccentricity and \(2 a\) is the length of the major axis. (b) The perihelion distance \(r_{\text {per }}\) and aphelion distance \(r_{\text {aph }}\) are defined as the minimum and maximum distances, respectively, of a planet from the sun. Show that \(r_{\text {per }}=a(1-e) \quad\) and \(\quad r_{\text {aph }}=a(1+e)\) (IMAGE CAN NOT COPY)

Show that $$x=a \cos t+h, \quad y=b \sin t+k ; \quad 0 \leq t \leq 2 \pi$$ are parametric equations of an ellipse with center \((h, k)\) and axes of lengths \(2 a\) and \(2 b\).

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$\frac{1}{3}(x+2)=y^{2}$$

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. $$r \cos \theta=5$$

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. $$\left\\{\begin{array}{l} x^{2}+4 y^{2}=20 \\ x+2 y=6 \end{array}\right.$$

Find an equation for the set of points in an xy-plane that are equidistant from the point \(P\) and the line \(L\) $$P(0,5) ; \quad k, y=-3$$

Show that $$\begin{array}{c}x=a \sec t+h, \quad y=b \tan t+k -\pi / 2

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$y^{2}=\frac{14}{3}-x^{2}$$

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. $$r \sin \theta=-2$$

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. $$\left\\{\begin{array}{l} 5 x^{2}+y^{2}=189 \\ 3 x+y=7 \end{array}\right.$$

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. Halley's Comet \(r_{\text {per }}=0.5871, \quad e=0.9673\) $$[-36,36,3] \text { by }[-24,24,3]$$

(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. $$y=x^{2}$$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+6 x-y^{2}=7$$

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. $$r=-3 \csc \theta$$

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. $$\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ 3 x^{2}+y^{2} &=43 \end{aligned}\right.$$

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 \(\quad r_{\text {per }}=1.251, \quad e=1.003\) $$[-18,18,3] \text { by }[-12,12,3]$$

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

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. $$r=-5$$

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 \(k\) $$F(3,0), \quad F(-3,0) ; \quad k=10$$