Chapter 10

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=\sqrt{t}, \quad y=t^{4}-1, \quad t \geq 0$$

Convert the rectangular coordinates to polar coordinates. $$(3,3 \sqrt{3})$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) endpoints of major and minor axes: (0,-7), (0,7),(-3,0),(3,0).

Sketch the graph of the equation and label the vertices. $$r=\frac{10}{4-3 \sin \theta}$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$y=3 x^{2}+x-4$$

Identify the conic whose equation is given and find its graph. If it is an ellipse, list its center, vertices, and foci. If it is a hyperbola, list its center, vertices, foci, and asymptotes. $$\frac{(y+3)^{2}}{25}-\frac{(x+1)^{2}}{16}=1$$

Convert the rectangular coordinates to polar coordinates. $$(-\sqrt{2}, \sqrt{6})$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) vertices (8,0) and (-8,0)\(;\) minor axis of length 8.

Sketch the graph of the equation and label the vertices. $$r=\frac{12}{3+4 \sin \theta}$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$y=-3 x^{2}+4 x-1$$

Identify the conic whose equation is given and find its graph. If it is an ellipse, list its center, vertices, and foci. If it is a hyperbola, list its center, vertices, foci, and asymptotes. $$\frac{(y+1)^{2}}{9}-\frac{(x-1)^{2}}{25}=1$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. \(x=e^{t}, \quad y=t, \quad\) any real number \(t\)

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$$

Convert the rectangular coordinates to polar coordinates. $$(2,4)$$

Sketch the graph of the equation and label the vertices. $$r=\frac{15}{3-2 \cos \theta}$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$y=-3 x^{2}+4 x+5$$

Identify the conic whose equation is given and find its graph. If it is an ellipse, list its center, vertices, and foci. If it is a hyperbola, list its center, vertices, foci, and asymptotes. $$\frac{(x+3)^{2}}{1}-\frac{(y-2)^{2}}{4}=1$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=2 e^{t}, \quad y=1-e^{t}, \quad t \geq 0$$

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$$

Convert the rectangular coordinates to polar coordinates. $$(3,-2)$$

Sketch the graph of the equation and label the vertices. $$r=\frac{32}{3+5 \sin \theta}$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$y=2 x^{2}-x-1$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=3 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi$$

Convert the rectangular coordinates to polar coordinates. $$(-5,2.5)$$

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$3 x^{2}+4 y^{2}=12$$

Sketch the graph of the equation and label the vertices. $$r=\frac{3}{1+\sin \theta}$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=4 \sin 2 t, \quad y=2 \cos 2 t, \quad 0 \leq t \leq 2 \pi$$

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$7 x^{2}+5 y^{2}=35$$

Convert the rectangular coordinates to polar coordinates. $$(-6.2,-3)$$

Sketch the graph of the equation and label the vertices. $$r=\frac{10}{3+2 \cos \theta}$$

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$y^{2}=x / 3$$

Identify the conic whose equation is given and find its graph. If it is an ellipse, list its center, vertices, and foci. If it is a hyperbola, list its center, vertices, foci, and asymptotes. $$(x-3)^{2}+12(y-2)^{2}=24$$

Sketch the graphs of the given curves and compare them. Do they differ and if so, how? (a) \(x=-4+6 t, \quad y=7-12 t, \quad 0 \leq t \leq 1\) (b) \(x=2-6 t, \quad y=-5+12 t, \quad 0 \leq t \leq 1\)

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$6 x^{2}+2 y^{2}=14$$

Convert the rectangular coordinates to polar coordinates. $$(0,-2)$$

Sketch the graph of the equation and label the vertices. $$r=\frac{10}{2+3 \sin \theta}$$

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$y^{2}=20 x$$

Identify the conic whose equation is given and find its graph. If it is an ellipse, list its center, vertices, and foci. If it is a hyperbola, list its center, vertices, foci, and asymptotes. $$4 y^{2}-x^{2}+6 x-24 y+11=0$$

Sketch the graphs of the given curves and compare them. Do they differ and if so, how? (a) \(x=t, \quad y=t^{2}\) (b) \(x=\sqrt{t}, \quad y=t\) (c) \(x=e^{t}, \quad y=e^{2 t}\)

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$5 x^{2}+y^{2}=5$$

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$y=2 x^{2}$$

Identify the conic whose equation is given and find its graph. If it is an ellipse, list its center, vertices, and foci. If it is a hyperbola, list its center, vertices, foci, and asymptotes. $$x^{2}-16 y^{2}=0$$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. circle with center (7,-4) and radius 6

Identify the conic section whose equation is given, and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$\frac{(x-1)^{2}}{4}+\frac{(y-5)^{2}}{9}=1$$

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$x^{2}-4 y=0$$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. circle with center ( 9,12 ) and radius 5

Identify the conic section whose equation is given, and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$\frac{(x-2)^{2}}{16}+\frac{(y+3)^{2}}{12}=1$$

Convert the rectangular coordinates to polar coordinates. $$(\sqrt{5}, \sqrt{10})$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Parabola; vertex \((2, \pi / 2)\)

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$y^{2}+12 x=0$$