Chapter 8

For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. \(-9 x^{2}+72 x+16 y^{2}+16 y+4=0\)

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. \(x^{2}+2 x+100 y^{2}-1000 y+2401=0\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{5}{5-11} \sin \theta\)

For the following exercises, determine the angle \(\theta\) that will eliminate the \(x y\) term and write the corresponding equation without the \(x y\) term. \(9 x^{2}-3 \sqrt{3} x y+6 y^{2}+4 y-3=0\)

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. \(5 x^{2}-50 x-4 y+113=0\)

For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. \(4 x^{2}+24 x-25 y^{2}+200 y-464=0\)

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. \(4 x^{2}+24 x+25 y^{2}+200 y+336=0\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r(5+2 \quad \cos \theta)=6\)

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

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

For the following exercises, determine the angle \(\theta\) that will eliminate the \(x y\) term and write the corresponding equation without the \(x y\) term. \(16 x^{2}+24 x y+9 y^{2}+6 x-6 y+2=0\)

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

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

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

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

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

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, 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, determine the angle \(\theta\) that will eliminate the \(x y\) term and write the corresponding equation without the \(x y\) term. \(4 x^{2}-2 \sqrt{3} x y+6 y^{2}-1=0\)

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

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

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

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. \(r=\frac{2}{3+3 \sin \theta}\)

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

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

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. \(r=\frac{10}{5-4 \sin \theta}\)

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

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(\frac{y^{2}}{9}-\frac{x^{2}}{25}=1\)

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

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. \(r=\frac{3}{1+2 \cos \theta}\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(y=-9 x^{2}\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(81 x^{2}-9 y^{2}=1\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(4 x^{2}+9 y^{2}=1\)

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. \(r=\frac{8}{4-5 \quad \cos \theta}\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \((y-2)^{2}=-\frac{4}{3}(x+2)\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(\frac{(y+5)^{2}}{9}-\frac{(x-4)^{2}}{25}=1\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(81 x^{2}+49 y^{2}=1\)