Chapter 8

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ y^{2}-24 x+4 y-68=0 $$

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. $$ 9 x^{2}+72 x+16 y^{2}+16 y+4=0 $$

For the following exercises, find the equations of the asymptotes for each hyperbola. $$ \frac{y^{2}}{3^{2}}-\frac{x^{2}}{3^{2}}=1 $$

Find the equations of the asymptotes for each hyperbola. \(\frac{y^{2}}{3^{2}}-\frac{x^{2}}{3^{2}}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(2-\cos \theta)=1 $$

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r(2-\cos \theta)=1 $$

Convert the polar equation of a conic section to a rectangular equation. $$ r(2-\cos \theta)=1 $$

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ x^{2}-4 x+2 y-6=0 $$

For the following exercises, find the foci for the given ellipses. $$ \frac{(x+3)^{2}}{25}+\frac{(y+1)^{2}}{36}=1 $$

For the following exercises, find the equations of the asymptotes for each hyperbola. $$ \frac{(x-3)^{2}}{5^{2}}-\frac{(y+4)^{2}}{2^{2}}=1 $$

Find the equations of the asymptotes for each hyperbola. \(\frac{(x-3)^{2}}{5^{2}}-\frac{(y+4)^{2}}{2^{2}}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(2.5-2.5 \sin \theta)=5 $$

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r(2.5-2.5 \sin \theta)=5 $$

Convert the polar equation of a conic section to a rectangular equation. $$ r(2.5-2.5 \sin \theta)=5 $$

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ y^{2}-6 y+12 x-3=0 $$

For the following exercises, find the foci for the given ellipses. $$ \frac{(x+1)^{2}}{100}+\frac{(y-2)^{2}}{4}=1 $$

For the following exercises, find the equations of the asymptotes for each hyperbola. $$ \frac{(y-3)^{2}}{3^{2}}-\frac{(x+5)^{2}}{6^{2}}=1 $$

For the following exercises, determine the angle ? that will eliminate the xy term and write the corresponding equation without the \(xy\) term. $$x^{2}+4 x y+4 y^{2}+3 x-2=0$$

Find the equations of the asymptotes for each hyperbola. \(\frac{(y-3)^{2}}{3^{2}}-\frac{(x+5)^{2}}{6^{2}}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{6 \sec \theta}{-2+3 \sec \theta} $$

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{6 \sec \theta}{-2+3 \sec \theta} $$

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{6 \sec \theta}{-2+3 \sec \theta} $$

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ 3 y^{2}-4 x-6 y+23=0 $$

For the following exercises, find the equations of the asymptotes for each hyperbola. $$ 9 x^{2}-18 x-16 y^{2}+32 y-151=0 $$

For the following exercises, determine the angle ? that will eliminate the xy term and write the corresponding equation without the \(xy\) term. $$x^{2}+4 x y+y^{2}-2 x+1=0$$

Find the equations of the asymptotes for each hyperbola. \(9 x^{2}-18 x-16 y^{2}+32 y-151=0\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{6 \csc \theta}{3+2 \csc \theta} $$

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{6 \csc \theta}{3+2 \csc \theta} $$

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{6 \csc \theta}{3+2 \csc \theta} $$

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ x^{2}+4 x+8 y-4=0 $$

For the following exercises, find the foci for the given ellipses. $$ x^{2}+4 y^{2}+4 x+8 y=1 $$

For the following exercises, find the equations of the asymptotes for each hyperbola. $$ 16 y^{2}+96 y-4 x^{2}+16 x+112=0 $$

Find the equations of the asymptotes for each hyperbola. \(16 y^{2}+96 y-4 x^{2}+16 x+112=0\)

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r=\frac{5}{2+\cos \theta} $$

Graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r=\frac{5}{2+\cos \theta} $$

For the following exercises, graph the parabola, labeling the focus and the directrix. $$ x=\frac{1}{8} y^{2} $$

For the following exercises, find the foci for the given ellipses. $$ 10 x^{2}+y^{2}+200 x=0 $$

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. $$ \frac{x^{2}}{49}-\frac{y^{2}}{16}=1 $$

Sketch a graph of the hyperbola, labeling vertices and foci. \(\frac{x^{2}}{49}-\frac{y^{2}}{16}=1\)

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r=\frac{2}{3+3 \sin \theta} $$

Graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r=\frac{2}{3+3 \sin \theta} $$

For the following exercises, graph the given ellipses, noting center, vertices, and foci. $$ \frac{x^{2}}{25}+\frac{y^{2}}{36}=1 $$

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. $$ \frac{x^{2}}{64}-\frac{y^{2}}{4}=1 $$

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation. $$x=y^{2}, \theta=45^{\circ}$$

For the following exercises, graph the parabola, labeling the focus and the directrix. $$y=36 x^{2}$$

Sketch a graph of the hyperbola, labeling vertices and foci. \(\frac{x^{2}}{64}-\frac{y^{2}}{4}=1\)

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r=\frac{10}{5-4 \sin \theta} $$

Graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r=\frac{10}{5-4 \sin \theta} $$

For the following exercises, graph the parabola, labeling the focus and the directrix $$ y=\frac{1}{36} x^{2} $$

For the following exercises, graph the given ellipses, noting center, vertices, and foci. $$ \frac{x^{2}}{16}+\frac{y^{2}}{9}=1 $$