Chapter 10

What conic does the polar equation \(r=a \sin \theta+b \cos \theta\) represent?

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y=\frac{1}{4}\left(x^{2}-2 x+5\right)$$

Factor the polynomial completely. \(x^{3}-16 x\)

Find the sum. $$\sum_{n=1}^{10} 4\left(\frac{3}{4}\right)^{n-1}$$

Convert the polar equation to rectangular form. $$r=\frac{2}{1+\sin \theta}$$

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$x=\frac{1}{4}\left(y^{2}+2 y+33\right)$$

Factor the polynomial completely. \(x^{2}+14 x+49\)

Convert the polar equation to rectangular form. $$r=\frac{6}{2-3 \sin \theta}$$

In your own words, define the term eccentricity and explain how it can be used to classify conics. Then explain how you can use the values of \(b\) and \(c\) to determine whether a polar equation of the form $$r=\frac{a}{b+c \sin \theta}$$ represents an ellipse, a parabola, or a hyperbola.

Factor the polynomial completely. \(2 x^{3}-24 x^{2}+72 x\)

Convert the polar equation to rectangular form. $$r=\frac{6}{2 \cos \theta-3 \sin \theta}$$

Find the exact value of the trigonometric expression when \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$\cos (u+v)$$

Factor the polynomial completely. \(6 x^{3}-11 x^{2}-10 x\)

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=6$$

Find the exact value of the trigonometric expression when \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$\sin (u+v)$$

Factor the polynomial completely. \(16 x^{3}+54\)

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=8$$

Find the exact value of the trigonometric expression when \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$\sin (u-v)$$

Factor the polynomial completely. \(4-x+4 x^{2}-x^{3}\)

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$\theta=\frac{\pi}{4}$$

Find the exact value of the trigonometric expression when \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$\cos (u-v)$$

Find the standard form of the equation of the parabola with the given characteristics. $$\text { Vertex: }(-2,0) ; \text { focus: }\left(-\frac{3}{2}, 0\right)$$

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$\theta=\frac{7 \pi}{6}$$

Find the standard form of the equation of the parabola with the given characteristics. $$\text { Vertex: }(3,-3) ; \text { focus: }\left(3,-\frac{9}{4}\right)$$

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=3 \sec \theta$$

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (5,2)\(;\) focus: (3,2)

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=2 \csc \theta$$

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (-1,2)\(;\) focus: (-1,0)

The center of a Ferris wheel lies at the pole of the polar coordinate system, where the distances are in feet. Passengers enter a car at \((30,-\pi / 2) .\) It takes 45 seconds for the wheel to complete one clockwise revolution. (a) Write a polar equation that models the possible positions of a passenger car. (b) Passengers enter a car. Find and interpret their coordinates after 15 seconds of rotation. (c) Convert the point in part (b) to rectangular coordinates. Interpret the coordinates.

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (0,4)\(;\) directrix: \(y=2\)

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (-2,1)\(;\) directrix: \(x=1\)

Determine whether the statement is true or false. Justify your answer. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\)

Find the standard form of the equation of the parabola with the given characteristics. Focus: (2,2)\(;\) directrix: \(x=-2\)

Determine whether the statement is true or false. Justify your answer. If \(\left(r, \theta_{1}\right)\) and \(\left(r, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\theta_{1}=\theta_{2}+2 \pi n\) for some integer \(n\).

Find the standard form of the equation of the parabola with the given characteristics. Focus: (0,0)\(;\) directrix: \(y=8\)

(a) Show that the distance between the points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is \(\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}\) (b) Simplify the Distance Formula for \(\theta_{1}=\theta_{2} .\) Is the simplification what you expected? Explain. (c) Simplify the Distance Formula for \(\theta_{1}-\theta_{2}=90^{\circ}\) Is the simplification what you expected? Explain.

In the rectangular coordinate system, each point \((x, y)\) has a unique representation. Explain why this is not true for a point \((r, \theta)\) in the polar coordinate system.

Find an equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,(4,8)$$

Convert the polar equation \(r=\cos \theta+3 \sin \theta\) to rectangular form and identify the graph.

Find an equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,\left(-3, \frac{2}{2}\right)$$

Convert the polar equation \(r=2(h \cos \theta+k \sin \theta)\) to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.

Find an equation of the tangent line to the parabola at the given point. $$x=-2 y^{2},(-2,-1)$$

Use the Law of sines or the Law of cosines to solve the triangle. $$a=13, b=19, c=25$$

Find an equation of the tangent line to the parabola at the given point. $$x=-2 y^{2},(-8,2)$$

Use the Law of sines or the Law of cosines to solve the triangle. $$A=24^{\circ}, a=10, b=6$$

Use the Law of sines or the Law of cosines to solve the triangle. $$A=56^{\circ}, C=38^{\circ}, c=12$$

Use the Law of sines or the Law of cosines to solve the triangle. $$B=71^{\circ}, a=21, c=29$$

Water is flowing from a horizontal pipe 48 feet above the ground. The falling stream of water has the shape of a parabola whose vertex (0,48) is at the end of the pipe (see figure). The stream of water strikes the ocean at the point \((10 \sqrt{3}, 0) .\) Find the equation of the path taken by the Water.

A cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cable touches the roadway midway between the towers. (a) Draw a sketch of the cable. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cable. (c) Complete the table by finding the height \(y\) of the suspension cable over the roadway at a distance of \(x\) meters from the center of the bridge. $$\begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 200 & 400 & 500 & 600 \\ \hline y & & & & & \\ \hline \end{array}$$

Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides. (See figure.) (a) Find an equation of the parabola with its vertex at the origin that models the road surface. (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?