Chapter 10

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) x^{2}+y^{2}-2 x+2 y+1=0

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=a, a>0 $$

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 \((4,0) .\)

a parametric representation of a curve is given. $$ x=3 t, y=2 t ;-\infty

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$

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. $$ y^{2}=4 x $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) x^{2}+y^{2}+6 x-2 y+6=0

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=2 a \cos \theta, a>0 $$

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

a parametric representation of a curve is given. $$ x=2 t, y=3 t ;-\infty

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{x^{2}}{9}-\frac{y^{2}}{4}=1 $$

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. $$ y^{2}=-12 x $$

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=2+\cos \theta $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 9 x^{2}+4 y^{2}+72 x-16 y+124=0

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 \(\left(-1,-\frac{1}{2} \pi\right) .\)

a parametric representation of a curve is given. $$ x=3 t-1, y=t ; 0 \leq t \leq 4 $$

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{-x^{2}}{9}+\frac{y^{2}}{4}=1 $$

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. $$ x^{2}=-12 y $$

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=5+4 \cos \theta $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 16 x^{2}-9 y^{2}+192 x+90 y-495=0

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 \(\left(3,-\frac{33}{2} \pi\right) .\)

a parametric representation of a curve is given. $$ x=4 t-2, y=2 t ; 0 \leq t \leq 3 $$

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{-x^{2}}{9}+\frac{y^{2}}{4}=-1 $$

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. $$ x^{2}=-16 y $$

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=3-3 \sin \theta $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 9 x^{2}+4 y^{2}+72 x-16 y+160=0

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) \(\left(-\sqrt{2}, \frac{5}{2} \pi\right)\)

a parametric representation of a curve is given. $$ x=4-t, y=\sqrt{t} ; 0 \leq t \leq 4 $$

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{-x^{2}}{9}+\frac{y}{4}=0 $$

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. $$ y^{2}=x $$

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=3+3 \sin \theta $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 16 x^{2}+9 y^{2}+192 x+90 y+1000=0

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 (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{-x^{2}}{9}=\frac{y}{4} $$

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. $$ y^{2}+3 x=0 $$

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=a(1+\cos \theta), a>0 $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) y^{2}-5 x-4 y-6=0

a parametric representation of a curve is given. $$ x=\frac{1}{s}, y=s ; 1 \leq s<10 $$

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ 9 x^{2}+4 y^{2}=9 $$

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. $$ 6 y-2 x^{2}=0 $$

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r^{2}=6 \cos 2 \theta $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 4 x^{2}+4 y^{2}+8 x-28 y-11=0

a parametric representation of a curve is given. $$ x=s, y=\frac{1}{s} ; 1 \leq s \leq 10 $$

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ x^{2}-4 y^{2}=4 $$

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. $$ 3 x^{2}-9 y=0 $$

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r^{2}=9 \sin 2 \theta $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 3 x^{2}+3 y^{2}-6 x+12 y+60=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) \((0,0)\)

a parametric representation of a curve is given. $$ x=t^{3}-4 t, y=t^{2}-4 ;-3 \leq t \leq 3 $$

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