Chapter 10

Plot the points whose polar coordinates are \(\left(3, \frac{1}{3} \pi\right)\), \(\left(1, \frac{1}{2} \pi\right),\left(4, \frac{1}{3} \pi\right),(0, \pi),(1,4 \pi),\left(3, \frac{11}{7} \pi\right),\left(\frac{5}{3}, \frac{1}{2} \pi\right)\), and \((4,0)\).

Name the conic corresponding to the given equation. \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ x^{2}+y^{2}-2 x+2 y+1=0 $$

In Problems 1–10, sketch the graph of the given equation and find the area of the region bounded by it. $$ r=a, a>0 $$

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$y^{2}=4 x$$

Plot the points whose polar coordinates are \((3,2 \pi)\), \(\left(2, \frac{1}{2} \pi\right), \quad\left(4,-\frac{1}{3} \pi\right), \quad(0,0),(1,54 \pi), \quad\left(3,-\frac{1}{6} \pi\right), \quad\left(1, \frac{1}{2} \pi\right), \quad\) and \(\left(3,-\frac{3}{2} \pi\right)\).

Name the conic corresponding to the given equation. \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ x^{2}+y^{2}+6 x-2 y+6=0 $$

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$y^{2}=-12 x$$

Plot the points whose polar coordinates are \((3,2 \pi)\), \(\left(-2, \frac{1}{3} \pi\right),\left(-2,-\frac{1}{4} \pi\right),(-1,1),(1,-4 \pi),\left(\sqrt{3},-\frac{7}{6} \pi\right),\left(-2, \frac{1}{4} \pi\right)\), and \(\left(-1,-\frac{1}{2} \pi\right)\).

Name the conic corresponding to the given equation. \(\frac{-x^{2}}{9}+\frac{y^{2}}{4}=1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 9 x^{2}+4 y^{2}+72 x-16 y+124=0 $$

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$x^{2}=-12 y$$

Plot the points whose polar coordinates are \(\left(3, \frac{9}{4} \pi\right)\), \(\left(-2, \frac{1}{2} \pi\right), \quad\left(-2,-\frac{1}{3} \pi\right), \quad(-1,-1), \quad(1,-7 \pi), \quad\left(-3,-\frac{1}{6} \pi\right)\), \(\left(-2,-\frac{1}{2} \pi\right)\), and \(\left(3,-\frac{33}{2} \pi\right)\).

Name the conic corresponding to the given equation. \(\frac{-x^{2}}{9}+\frac{y^{2}}{4}=-1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 16 x^{2}-9 y^{2}+192 x+90 y-495=0 $$

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=4 t-2, y=2 t ; 0 \leq t \leq 3 $$

\(r=5+4 \cos \theta\)

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$x^{2}=-16 y$$

Plot the points whose polar coordinates follow. For each point, give four other pairs of polar coordinates, two with positive \(r\) and two with negative \(r\). (a) \(\left(1, \frac{1}{2} \pi\right)\) (b) \(\left(-1, \frac{1}{4} \pi\right)\) (c) \(\left(\sqrt{2},-\frac{1}{3} \pi\right)\) (d) \(\left(-\sqrt{2}, \frac{5}{2} \pi\right)\)

Name the conic corresponding to the given equation. \(\frac{-x^{2}}{9}+\frac{y}{4}=0\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 9 x^{2}+4 y^{2}+72 x-16 y+160=0 $$

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=4-t, y=\sqrt{t} ; 0 \leq t \leq 4 $$

\(r=3-3 \sin \theta\)

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$y^{2}=x$$

Plot the points whose polar coordinates follow. For each point, give four other pairs of polar coordinates, two with positive \(r\) and two with negative \(r\). (a) \(\left(3 \sqrt{2}, \frac{7}{2} \pi\right)\) (b) \(\left(-1, \frac{15}{4} \pi\right)\) (c) \(\left(-\sqrt{2},-\frac{2}{3} \pi\right)\) (d) \(\left(-2 \sqrt{2}, \frac{29}{2} \pi\right)\)

Name the conic corresponding to the given equation. \(\frac{-x^{2}}{9}=\frac{y}{4}\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 16 x^{2}+9 y^{2}+192 x+90 y+1000=0 $$

\(r=3+3 \sin \theta\)

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=4 \sin \theta\)

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$y^{2}+3 x=0$$

Name the conic corresponding to the given equation. \(9 x^{2}+4 y^{2}=9\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ y^{2}-5 x-4 y-6=0 $$

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$6 y-2 x^{2}=0$$

Name the conic corresponding to the given equation. \(x^{2}-4 y^{2}=4\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}+4 y^{2}+8 x-28 y-11=0 $$

\(r^{2}=6 \cos 2 \theta\)

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$3 x^{2}-9 y=0$$

Find polar coordinates of the points whose Cartesian coordinates are given. (a) \((3 \sqrt{3}, 3)\) (b) \((-2 \sqrt{3}, 2)\) (c) \((-\sqrt{2},-\sqrt{2})\) (d) \((0,0)\)

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 3 x^{2}+3 y^{2}-6 x+12 y+60=0 $$

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=t^{3}-4 t, y=t^{2}-4 ;-3 \leq t \leq 3 $$

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is at \((2,0)\)

Find polar coordinates of the points whose Cartesian coordinates are given. (a) \((-3 / \sqrt{3}, 1 / \sqrt{3})\) (b) \((-\sqrt{3} / 2, \sqrt{3} / 2)\) (c) \((0,-2)\) (d) \((3,-4)\)

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(\frac{x^{2}}{16}-\frac{y^{2}}{4}=1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}-4 y^{2}-2 x+2 y+1=0 $$

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Directrix is \(x=3\)

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x-3 y+2=0\)

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes. \(\frac{-x^{2}}{9}+\frac{y^{2}}{4}=1\)

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}-4 y^{2}+8 x+12 y-5=0 $$