Chapter 12

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 4 x^{2}+y^{2}=16 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ y^{2}=3 x $$

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

\(13-16\) . Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$ y^{2}-\frac{(x+1)^{2}}{4}=1 $$

\(15-28=(a)\) 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 eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 4 x^{2}+y^{2}=16 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ y=5 x^{2} $$

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

\(13-16\) . 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 eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 5 x^{2}+6 y^{2}=30 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ y=-2 x^{2} $$

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

A polar equation of a conic is given. (a) Show that the conic is a parabola and sketch its graph. (b) Find the vertex and directrix and indicate them on the graph. $$ r=\frac{4}{1-\sin \theta} $$

\(15-28=(a)\) 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. $$ x^{2}+2 \sqrt{3} x y-y^{2}+2=0 $$

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

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ x=-8 y^{2} $$

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

A polar equation of a conic is given. (a) Show that the conic is a parabola and sketch its graph. (b) Find the vertex and directrix and indicate them on the graph. $$ r=\frac{3}{2+2 \sin \theta} $$

\(15-28=(a)\) 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. $$ 13 x^{2}+6 \sqrt{3} x y+7 y^{2}=16 $$

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

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ x=\frac{1}{2} y^{2} $$

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

A polar equation of a conic is given. (a) Show that the conic is a parabola and sketch its graph. (b) Find the vertex and directrix and indicate them on the graph. $$ r=\frac{5}{3+3 \cos \theta} $$

\(15-28=(a)\) 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. $$ 11 x^{2}-24 x y+4 y^{2}+20=0 $$

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

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ x^{2}+6 y=0 $$

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

A polar equation of a conic is given. (a) Show that the conic is a parabola and sketch its graph. (b) Find the vertex and directrix and indicate them on the graph. $$ r=\frac{2}{5-5 \cos \theta} $$

\(15-28=(a)\) 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. $$ 21 x^{2}+10 \sqrt{3} x y+31 y^{2}=144 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ x-7 y^{2}=0 $$

A polar equation of a conic is given. (a) Show that the conic is an ellipse, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the ellipse and the lengths of the major and minor axes. $$ r=\frac{4}{2-\cos \theta} $$

\(15-28=(a)\) 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. $$ 21 x^{2}+10 \sqrt{3} x y+31 y^{2}=144 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ 5 x+3 y^{2}=0 $$

A polar equation of a conic is given. (a) Show that the conic is an ellipse, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the ellipse and the lengths of the major and minor axes. $$ r=\frac{6}{3-2 \sin \theta} $$

\(15-28=(a)\) 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. $$ 153 x^{2}+192 x y+97 y^{2}=225 $$

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

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ 8 x^{2}+12 y=0 $$

A polar equation of a conic is given. (a) Show that the conic is an ellipse, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the ellipse and the lengths of the major and minor axes. $$ r=\frac{12}{4+3 \sin \theta} $$

\(23-34\) 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 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. $$ y^{2}=4(x+2 y) $$

\(15-28=(a)\) 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. $$ x^{2}+2 x y+y^{2}+x-y=0 $$

\(23-28\) Use a graphing device to graph the parabola. $$ x^{2}=16 y $$

A polar equation of a conic is given. (a) Show that the conic is an ellipse, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the ellipse and the lengths of the major and minor axes. $$ r=\frac{18}{4+3 \cos \theta} $$

\(23-34\) 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 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. $$ 9 x^{2}-36 x+4 y^{2}=0 $$

\(15-28=(a)\) 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. $$ 25 x^{2}-120 x y+144 y^{2}-156 x-65 y=0 $$

\(23-28\) Use a graphing device to graph the parabola. $$ x^{2}=-8 y $$

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices anddirectrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the asymptotes. $$ r=\frac{8}{1+2 \cos \theta} $$

\(23-34\) 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 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. $$ x^{2}-4 y^{2}-2 x+16 y=20 $$

\(15-28=(a)\) 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. $$ 2 \sqrt{3} x^{2}-6 x y+\sqrt{3} x+3 y=0 $$

\(23-28\) Use a graphing device to graph the parabola. $$ y^{2}=-\frac{1}{3} x $$

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices anddirectrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the asymptotes. $$ r=\frac{10}{1-4 \sin \theta} $$