Chapter 10

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=1+\cos \theta $$

Find the areas of the regions. Inside the oval limaçon \(r=4+2 \cos \theta\)

Which polar coordinate pairs label the same point? $$ \begin{array}{lll}{\text { a. }(3,0)} & {\text { b. }(-3,0)} & {\text { c. }(2,2 \pi / 3)} \\ {\text { d. }(2,7 \pi / 3)} & {\text { e. }(-3, \pi)} & {\text { f. }(2, \pi / 3)} \\ {\text { g. }(-3,2 \pi)} & {\text { h. }(-2,-\pi / 3)}\end{array} $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=\cos t, \quad y=\sin t, \quad 0 \leq t \leq \pi\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 16 x^{2}+25 y^{2}=400 $$

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(x^{2}-3 x y+y^{2}-x=0\)

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=2-2 \cos \theta $$

Which polar coordinate pairs label the same point? $$ \begin{array}{lll}{\text { a. }(-2, \pi / 3)} & {\text { b. }(2,-\pi / 3)} & {\text { c. }(r, \theta)} \\ {\text { d. }(r, \theta+\pi)} & {\text { e. }(-r, \theta)} & {\text { f. }(2,-2 \pi / 3)} \\ {\text { g. }(-r, \theta+\pi)} & {\text { h. }(-2,2 \pi / 3)}\end{array} $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=\sin (2 \pi(1-t)), \quad y=\cos (2 \pi(1-t)) ; \quad 0 \leq t \leq 1\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 7 x^{2}+16 y^{2}=112 $$

Find the areas of the regions. Inside the cardioid \(r=a(1+\cos \theta), \quad a>0\)

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(3 x^{2}-18 x y+27 y^{2}-5 x+7 y=-4\)

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=1-\sin \theta $$

Find the areas of the regions. Inside one leaf of the four-leaved rose \(r=\cos 2 \theta\)

Plot the following points (given in polar coordinates). Then find all the polar coordinates of each point. $$ \begin{array}{ll}{\text { a. }(2, \pi / 2)} & {\text { b. }(2,0)} \\ {\text { c. }(-2, \pi / 2)} & {\text { d. }(-2,0)}\end{array} $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=4 \cos t, \quad y=5 \sin t ; \quad 0 \leq t \leq \pi\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 2 x^{2}+y^{2}=2 $$

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(3 x^{2}-7 x y+\sqrt{17} y^{2}=1\)

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=1+\sin \theta $$

Plot the following points (given in polar coordinates). Then find all the polar coordinates of each point. $$ \begin{array}{ll}{\text { a. }(3, \pi / 4)} & {\text { b. }(-3, \pi / 4)} \\\ {\text { c. }(3,-\pi / 4)} & {\text { d. }(-3,-\pi / 4)}\end{array} $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=4 \sin t, \quad y=5 \cos t ; \quad 0 \leq t \leq 2 \pi\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 2 x^{2}+y^{2}=4 $$

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(2 x^{2}-\sqrt{15} x y+2 y^{2}+x+y=0\)

Find the areas of the regions. Inside the lemniscate \(r^{2}=2 a^{2} \cos 2 \theta, \quad a>0\)

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=2+\sin \theta $$

Find the areas of the regions. Inside one loop of the lemniscate \(r^{2}=4 \sin 2 \theta\)

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=t, \quad y=\sqrt{t} ; \quad t \geq 0\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 3 x^{2}+2 y^{2}=6 $$

Sketch the lines in Exercises \(5-8\) and find Cartesian equations for them. $$ r \cos \left(\theta-\frac{\pi}{4}\right)=\sqrt{2} $$

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(x^{2}+2 x y+y^{2}+2 x-y+2=0\)

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=1+2 \sin \theta $$

Find the Cartesian coordinates of the following points (given in polar coordinates). $$ \begin{array}{ll}{\text { a. }(\sqrt{2}, \pi / 4)} & {\text { b. }(1,0)} \\\ {\text { c. }(0, \pi / 2)} & {\text { d. }(-\sqrt{2}, \pi / 4)} \\ {\text { e. }(-3,5 \pi / 6)} & {\text { f. }\left(5, \tan ^{-1}(4 / 3)\right)} \\\ {\text { g. }(-1,7 \pi)} & {\text { h. }(2 \sqrt{3}, 2 \pi / 3)}\end{array} $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=\sec ^{2} t-1, \quad y=\tan t ; \quad-\pi / 2< t<\pi / 2\)

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(2 x^{2}-y^{2}+4 x y-2 x+3 y=6\)

Find the areas of the regions Inside the six-leaved rose \(r^{2}=2 \sin 3 \theta\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 9 x^{2}+10 y^{2}=90 $$

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=\sin (\theta / 2) $$

Find the areas of the regions Shared by the circles \(r=2 \cos \theta\) and \(r=2 \sin \theta\)

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(7-22\) . $$ r=2 $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=-\sec t, \quad y=\tan t ; \quad-\pi / 2< t<\pi / 2\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 6 x^{2}+9 y^{2}=54 $$

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(x^{2}+4 x y+4 y^{2}-3 x=6\)

Match each conic section in Exercises \(5-8\) with one of these equations: $$\begin{array}{ll}{\frac{x^{2}}{4}+\frac{y^{2}}{9}=1,} & {\frac{x^{2}}{2}+y^{2}=1} \\ {\frac{y^{2}}{4}-x^{2}=1,} & {\frac{x^{2}}{4}-\frac{y^{2}}{9}=1}\end{array}$$ Then find the conic section's foci and vertices. If the conic section is a hyperbola, find its asymptotes as well. (GRAPH NOT COPY)

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=\cos (\theta / 2) $$

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=\csc t, \quad y=\cot t ; \quad 0< t<\pi\)

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 169 x^{2}+25 y^{2}=4225 $$

Find the areas of the regions Shared by the circles \(r=1\) and \(r=2 \sin \theta\)

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r^{2}=\cos \theta $$

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ y^{2}=12 x $$

Find the areas of the regions Shared by the circle \(r=2\) and the cardioid \(r=2(1-\cos \theta)\)