Chapter 10

Find an equation for the set of points in an xy-plane such that the difference of the distances from \(F\) and \(F\) is \(k\) $$F(0,10), \quad F(0,-10) ; \quad k=16$$

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. $$r(\sin \theta-2 \cos \theta)=6$$

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. $$x=-\frac{1}{3} \sqrt{9-y^{2}}$$

Find an equation for the parabola that has a vertical axis and passes through the given points. $$P(2,5), \quad Q(-2,-3), \quad R(1,6)$$

Find an equation for the set of points in an xy-plane such that the difference of the distances from \(F\) and \(F\) is \(k\) $$F(0,17), \quad F(0,-17) ; \quad k=30$$

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. $$r(3 \cos \theta-4 \sin \theta)=12$$

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. $$x=\frac{4}{5} \sqrt{25-y^{2}}$$

Find an equation for the parabola that has a vertical axis and passes through the given points. $$P(3,-1), Q(1,-7), \quad R(-2,14)$$

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. $$r\left(\sin \theta+r \cos ^{2} \theta\right)=1$$

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. $$x=1+2 \sqrt{1-\frac{(y+2)^{2}}{9}}$$

Find an equation for the parabola that has a horizontal axis and passes through the given points. $$P(-1,1), Q(11,-2), \quad R(5,-1)$$

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. $$r\left(r \sin ^{2} \theta-\cos \theta\right)=3$$

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. $$x=-2-5 \sqrt{1-\frac{(y-1)^{2}}{16}}$$

Find an equation for the parabola that has a horizontal axis and passes through the given points. $$P(2,1), \quad Q(6,2), \quad R(12,-1)$$

Graph the curve. $$x=3 \sin ^{5} t, \quad y=3 \cos ^{5} t, \quad 0 \leq t \leq 2 \pi$$

Find an equation for the indicated part of the hyperbola. Lower branch of \(\frac{y^{2}}{25}-\frac{x^{2}}{36}=1\)

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. $$r=8 \sin \theta-2 \cos \theta$$

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. $$y=2-7 \sqrt{1-\frac{(x+1)^{2}}{9}}$$

A mirror for a reflecting telescope has the shape of a (finite) paraboloid of diameter 8 inches and depth 1 inch. How far from the center of the mirror will the incoming light collect? (IMAGE CAN'T COPY)

Graph the curve. $$\begin{aligned}x=8 \cos t-2 \cos 4 t & y &=8 \sin t-2 \sin 4 t ; \quad 0 \leq t \leq 2 \pi\end{aligned}$$

Find an equation for the indicated part of the hyperbola. Upper branch of \(\frac{y^{2}}{49}-\frac{x^{2}}{25}=1\)

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. $$r=2 \cos \theta-4 \sin \theta$$

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. $$y=-1+\sqrt{1-\frac{(x-3)^{2}}{16}}$$

A satellite antenna dish has the shape of a paraboloid that is 10 feet across at the open end and is 3 feet deep. At what distance from the center of the dish should the receiver be placed to receive the greatest intensity of sound waves?

Graph the curve. $$x=3 t-2 \sin t, \quad y=3-2 \cos t, \quad-8 \leq t \leq 8$$

Find an equation for the indicated part of the hyperbola. Left branch of \(\frac{x^{2}}{4}-\frac{y^{2}}{16}=1\)

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. $$r=\tan \theta$$

Dimensions of an arch An arch of a bridge is semielliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway, as shown in the figure. Find the height of the arch 6 feet from the center of the base. (GRAPH CANT COPY)

A searchlight reflector has the shape of a paraboloid, with the light source at the focus. If the reflector is 3 feet across at the opening and 1 foot deep. where is the focus?

Graph the curve. $$x=2 t-3 \sin t, \quad y=2-3 \cos t, \quad-8 \leq t \leq 8$$

Find an equation for the indicated part of the hyperbola. Right branch of \(\frac{x^{2}}{16}-\frac{y^{2}}{4}=1\)

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. $$r=6 \cot \theta$$

A flashlight mirror has the shape of a paraboloid of diameter 4 inches and depth \(\frac{3}{4}\) inch, as shown in the figure. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?(IMAGE CAN'T COPY)

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure. $$\begin{aligned} &C_{1}: x=2 \sin 3 t, \quad y=3 \cos 2 t, \quad-\pi / 2 \leq t \leq \pi / 2\\\ &C_{2}: x=\frac{1}{4} \cos t+\frac{3}{4}, \quad y=\frac{1}{4} \sin t+\frac{3}{2} ; \quad 0 \leq t \leq 2 \pi\\\ &C_{3}: x=\frac{1}{4} \cos t-\frac{3}{4}, \quad y=\frac{1}{4} \sin t+\frac{3}{2} ; \quad 0 \leq t \leq 2 \pi\\\ &C_{4}: x=\frac{3}{4} \cos t, \quad y=\frac{1}{4} \sin t, \quad 0 \leq t \leq 2 \pi\\\ &C_{5}: x=\frac{1}{4} \cos t, \quad y=\frac{1}{8} \sin t+\frac{3}{4} ; \quad \pi \leq t \leq 2 \pi \end{aligned}$$

Find an equation for the indicated part of the hyperbola. Right halves of the branches of \(\frac{y^{2}}{4}-\frac{x^{2}}{81}=1\)

Sketch the graph of the polar equation. $$r=5$$

Earth's orbit Assume that the length of the major axis of Earth's orbit is \(186,000,000\) miles and that the eccentricity is 0.017. Approximate, to the nearest 1000 miles, the maximum and minimum distances between Earth and the sun.

A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 5 inches from the vertex. Determine the width of the dish if the depth is to be 2 feet.

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure. $$\begin{aligned}&C_{1}: x=\frac{3}{2} \cos t+1, y=\sin t-1 ; \quad-\pi / 2 \leq t \leq \pi / 2\\\&C_{2}: x=\frac{3}{2} \cos t+1, y=\sin t+1 ; \quad-\pi / 2 \leq t \leq \pi / 2\\\&C_{3}: x=1, \quad y=2 \tan t, \quad-\pi / 4 \leq t \leq \pi / 4\end{aligned}$$

Find an equation for the indicated part of the hyperbola. Left halves of the branches of \(\frac{y^{2}}{36}-\frac{x^{2}}{16}=1\)

Sketch the graph of the polar equation. $$r=-2$$

Mercury's orbit The planet Mercury travels in an elliptical orbit that has eccentricity 0.206 and major axis of length 0.774 AU. Find the maximum and minimum distances between Mercury and the sun.

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure. $$\begin{array}{lll}C_{1}: x=\tan t, & y=3 \tan t ; & 0 \leq t \leq \pi / 4 \\\C_{2}: x=1+\tan t, & y=3-3 \tan t ; & 0 \leq t \leq \pi / 4 \\\C_{3}: x=\frac{1}{2}+\tan t, & y=\frac{3}{2} ; & 0 \leq t \leq \pi / 4\end{array}$$

Find an equation for the indicated part of the hyperbola. Upper halves of the branches of \(\frac{x^{2}}{9}-\frac{y^{2}}{36}=1\)

Sketch the graph of the polar equation. $$\theta=-\pi / 6$$

Parabolic reflector (a) The focal length of the (finite) paraboloid in the figure is the distance \(p\) between its vertex and focus. Express \(p\) in terms of \(r\) and \(h\) (b) \(\mathrm{A}\) reflector is to be constructed with a focal length of 10 feet and a depth of 5 feet. Find the radius of the reflector. (IMAGE CAN'T COPY)

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure. $$\begin{array}{lll}C_{1}: x=1+\cos t, & y=1+\sin t, & \pi / 3 \leq t \leq 2 \pi \\\C_{2}: x=1+\tan t, & y=1 ; & 0 \leq t \leq \pi / 4\end{array}$$

Find an equation for the indicated part of the hyperbola. Lower halves of the branches of \(\frac{x^{2}}{9}-\frac{y^{2}}{49}=1\)

Sketch the graph of the polar equation. $$\theta=\pi / 4$$

Sketch the graph of the polar equation. $$r=3 \cos \theta$$