Chapter 10

Find an equation for the indicated half of the parabola. Lower half of \((y+1)^{2}=x+3\)

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 \(1973.99 \quad r_{\text {per }}=0.142, \quad e=1.000\) $$[-18,18,3] \text { by }[-12,12,3]$$

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

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. $$r=2$$

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 \(k\) $$F(12,0), \quad F(-12,0) ; \quad k=26$$

Find an equation for the indicated half of the parabola. Upper half of \((y-2)^{2}=x-4\)

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$4 x^{2}-16 x+9 y^{2}+36 y=-16$$

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. $$r-6 \sin \theta=0$$

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 \(k\) $$F(0,15), \quad F(0,-15) ; \quad k=34$$

Find an equation for the indicated half of the parabola. Right half of \((x+1)^{2}=y-4\)

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

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. $$r-6 \cos \theta=0$$

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 \(k\) $$F(0,8), \quad F(0,-8) ; \quad k=20$$

Find an equation for the indicated half of the parabola. Left half of \((x+3)^{2}=y+2\)

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+3 x=3 y-6$$

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. $$\theta=\pi / 4$$

Find an equation for the indicated half of the parabola. Upper half of \((y-5)^{2}=x+2\)

Shown in the figure is the Lissajous figure given by $$x=2 \sin 3 t, \quad y=3 \sin 1.5 t, \quad t \geq 0$$ Find the period of the figure-that is, the length of the smallest \(t\) -interval that traces the curve. CAN'T COPY THE GRAPH

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$9 x^{2}-y^{2}=10-2 y$$

Find an equation for the indicated half of the parabola. Lower half of \((y+4)^{2}=x-3\)

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. $$\left\\{\begin{aligned}y^{2}-4 x^{2} &=16 \\\y-x &=4\end{aligned}\right.$$

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. $$r^{2}\left(4 \sin ^{2} \theta-9 \cos ^{2} \theta\right)=36$$

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. $$\text { Left half of } \frac{x^{2}}{36}+\frac{y^{2}}{25}=1$$

Find an equation for the indicated half of the parabola. Left half of \((x-2)^{2}=y+1\)

Lissajous figures are used in the study of electrical circuits to determine the phase difference \(\phi\) between a known voltage \(V_{1}(t)=A \sin (\omega t)\) and an unknown voltage \(V_{2}(\vec{t})=B \sin (\omega t+\phi)\) having the same frequency. The voltages are graphed parametrically as \(x=V_{1}(t)\) and \(y=V_{2}(t)\) If \(\phi\) is acute, then $$\phi=\sin ^{-1} \frac{y_{\mathrm{int}}}{y_{\max }}$$ where \(y_{\text {int }}\) is the nonnegative \(y\) -intercept and \(y_{\max }\) is the maximum \(y\) -value on the curve. (a) Graph the parametric curve \(x=V_{1}(t)\) and \(y=V_{2}(t)\) for the specified range of \(t\) (b) Use the graph to approximate \(\phi\) in degrees. $$\begin{aligned}&V_{1}(t)=6 \sin (120 \pi t), \quad V_{2}(t)=5 \cos (120 \pi t)&0 \leq t \leq 0.02\end{aligned}$$

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. $$\left\\{\begin{array}{l}x^{2}-y^{2}=4 \\\y^{2}-3 x=0\end{array}\right.$$

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. $$r^{2}\left(\cos ^{2} \theta+4 \sin ^{2} \theta\right)=16$$

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. $$\text { Right half of } \frac{x^{2}}{9}+\frac{y^{2}}{121}=1$$

Find an equation for the indicated half of the parabola. Right half of \((x-4)^{2}=y-5\)

Lissajous figures are used in the study of electrical circuits to determine the phase difference \(\phi\) between a known voltage \(V_{1}(t)=A \sin (\omega t)\) and an unknown voltage \(V_{2}(\vec{t})=B \sin (\omega t+\phi)\) having the same frequency. The voltages are graphed parametrically as \(x=V_{1}(t)\) and \(y=V_{2}(t)\) If \(\phi\) is acute, then $$\phi=\sin ^{-1} \frac{y_{\mathrm{int}}}{y_{\max }}$$ where \(y_{\text {int }}\) is the nonnegative \(y\) -intercept and \(y_{\max }\) is the maximum \(y\) -value on the curve. (a) Graph the parametric curve \(x=V_{1}(t)\) and \(y=V_{2}(t)\) for the specified range of \(t\) (b) Use the graph to approximate \(\phi\) in degrees. $$\begin{aligned}V_{1}(t)=80 \sin (60 \pi t), & V_{2}(t)=70 \cos (60 \pi t-\pi / 3) & 0 \leq t \leq 0.035\end{aligned}$$

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. $$\left\\{\begin{array}{l}x^{2}-3 y^{2}=4 \\\x^{2}+4 y^{2}=32\end{array}\right.$$

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. $$r^{2} \sin 2 \theta=4$$

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. Upper half of \(x^{2}+3 y^{2}=17\)

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. $$y=\sqrt{x-6}-2$$

Lissajous figures are used in the study of electrical circuits to determine the phase difference \(\phi\) between a known voltage \(V_{1}(t)=A \sin (\omega t)\) and an unknown voltage \(V_{2}(\vec{t})=B \sin (\omega t+\phi)\) having the same frequency. The voltages are graphed parametrically as \(x=V_{1}(t)\) and \(y=V_{2}(t)\) If \(\phi\) is acute, then $$\phi=\sin ^{-1} \frac{y_{\mathrm{int}}}{y_{\max }}$$ where \(y_{\text {int }}\) is the nonnegative \(y\) -intercept and \(y_{\max }\) is the maximum \(y\) -value on the curve. (a) Graph the parametric curve \(x=V_{1}(t)\) and \(y=V_{2}(t)\) for the specified range of \(t\) (b) Use the graph to approximate \(\phi\) in degrees. $$\begin{aligned}&V_{1}(t)=163 \sin (120 \pi t), \quad V_{2}(t)=163 \sin (120 \pi t+\pi / 4)&-0 \leq t \leq 0.02\end{aligned}$$

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. $$\left\\{\begin{aligned}y^{2}-3 x^{2} &=6 \\\x^{2}+y^{2} &=106\end{aligned}\right.$$

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. $$r^{2} \sin 2 \theta=-10$$

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. Lower half of \(2 x^{2}+5 y^{2}=12\)

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. $$y=-\sqrt{x+3}+4$$

Graph the Lissajous figure in the viewing rectangle \([-1,1]\) by \([-1,1]\) for the specified range of \(t\). $$x(t)=\sin (6 \pi t), \quad y(t)=\cos (5 \pi t) ; \quad 0 \leq t \leq 2$$

Find an equation for the set of points in an xy-plane such that the difference of the distances from \(F\) and \(F\) is \(k\) $$F(13,0), \quad F(-13,0) ; \quad k=24$$

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. $$r^{2} \cos 2 \theta=1$$

Exer. \(51-58:\) Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$y=11 \sqrt{1-\frac{x^{2}}{49}}$$

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. $$x=-\sqrt{y+7}-3$$

Graph the Lissajous figure in the viewing rectangle \([-1,1]\) by \([-1,1]\) for the specified range of \(t\). $$x(t)=\sin (4 t), \quad y(t)=\sin (3 t+\pi / 6) ; \quad 0 \leq t \leq 6.5$$

Find an equation for the set of points in an xy-plane such that the difference of the distances from \(F\) and \(F\) is \(k\) $$F(5,0), \quad F(-5,0) ; \quad k=8$$

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. $$r^{2} \cos 2 \theta=-9$$

Exer. \(51-58:\) Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$y=-6 \sqrt{1-\frac{x^{2}}{25}}$$

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. $$x=\sqrt{y-4}+8$$