Chapter 11

Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$ (x-8)^{2}-(y+6)^{2}=1 $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(9 x^{2}-4 y^{2}=36\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$y^{2}=3 x$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=2 \sin t, \quad y=2 \cos t, \quad 0 \leq t \leq \pi $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 2 x^{2}+y^{2}=3 $$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}+2 \sqrt{3} x y-y^{2}=4, \quad \phi=30^{\circ}$$

Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$ y^{2}-\frac{(x+1)^{2}}{4}=1 $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(25 y^{2}-9 x^{2}=225\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$y=5 x^{2}$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 5 x^{2}+6 y^{2}=30 $$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x y=x+y, \quad \phi=\pi / 4$$

Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$ \frac{(y-1)^{2}}{25}-(x+3)^{2}=1 $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(x^{2}-y^{2}+4=0\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$y=-2 x^{2}$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ x^{2}+4 y^{2}=1 $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$x y=8$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(x^{2}-4 y^{2}-8=0\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=-8 y^{2}$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\sin ^{2} t, \quad y=\cos t $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 9 x^{2}+4 y^{2}=1 $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$x y+4=0$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(x^{2}-2 y^{2}=3\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=\frac{1}{2} y^{2}$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cos t, \quad y=\cos 2 t $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ \frac{1}{2} x^{2}+\frac{1}{8} y^{2}=\frac{1}{4} $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$x^{2}+2 x y+y^{2}+x-y=0$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{4}{1+3 \cos \theta}$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(4 y^{2}-x^{2}=1\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x^{2}+6 y=0$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cos 2 t, \quad y=\sin 2 t $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}=16$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{8}{3+3 \cos \theta}$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(9 x^{2}-16 y^{2}=1\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x-7 y^{2}=0$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\sec t, \quad y=\tan t, \quad 0 \leq t<\pi / 2 $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$x^{2}+2 \sqrt{3} x y-y^{2}+2=0$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{2}{1-\cos \theta}$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$5 x+3 y^{2}=0$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cot t, \quad y=\csc t, \quad 0

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 20 x^{2}+4 y^{2}=5 $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$21 x^{2}+10 \sqrt{3} x y+31 y^{2}=144$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{10}{3-2 \sin \theta}$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$8 x^{2}+12 y=0$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\tan t, \quad y=\cot t, \quad 0

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$11 x^{2}-24 x y+4 y^{2}+20=0$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{6}{2+\sin \theta}$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 9 x^{2}-36 x+4 y^{2}=0 $$

Use a graphing device to graph the parabola. $$x^{2}=16 y$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\sec t, \quad y=\tan ^{2} t, \quad 0 \leq t<\pi / 2 $$