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. \(9 y^{2}-4 x^{2}=1\)

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. \(4 x^{2}+16 y^{2}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r(3+5 \sin \theta)=11\)

For the following exercises, determine which conic section is represented based on the given equation. \(2 x^{2}+4 \sqrt{3} x y+6 y^{2}-6 x-3=0\)

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. \(x=36 y^{2}\)

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. \(\frac{(x-1)^{2}}{9}-\frac{(y-2)^{2}}{16}=1\)

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. \(\frac{(x-2)^{2}}{49}+\frac{(y-4)^{2}}{25}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r(4-5 \sin \theta)=1\)

For the following exercises, determine which conic section is represented based on the given equation. \(-x^{2}+4 \sqrt{2} x y+2 y^{2}-2 y+1=0\)

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. \(x=\frac{1}{36} y^{2}\)

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. \(\frac{(y-6)^{2}}{36}-\frac{(x+1)^{2}}{16}=1\)

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. \(\frac{(x-2)^{2}}{81}+\frac{(y+1)^{2}}{16}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r(7+8 \cos \theta)=7\)

For the following exercises, determine which conic section is represented based on the given equation. \(8 x^{2}+4 \sqrt{2} x y+4 y^{2}-10 x+1=0\)

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. \((x-1)^{2}=4(y-1)\)

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. \(\frac{(x-2)^{2}}{49}-\frac{(y+7)^{2}}{49}=1\)

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. \(\frac{(x+5)^{2}}{4}+\frac{(y-7)^{2}}{9}=1\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{4}{1+3 \sin \theta}\)

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(3 x^{2}+x y+3 y^{2}-5=0, \theta=45^{\circ}\)

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. \((y-2)^{2}=\frac{4}{5}(x+4)\)

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. \(4 x^{2}-8 x-9 y^{2}-72 y+112=0\)

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. \(\frac{(x-7)^{2}}{49}+\frac{(y-7)^{2}}{49}=1\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{2}{5-3 \sin \theta}\)

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(4 x^{2}-x y+4 y^{2}-2=0, \theta=45^{\circ}\)

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. \((y-4)^{2}=2(x+3)\)

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. \(-9 x^{2}-54 x+9 y^{2}-54 y+81=0\)

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. \(4 x^{2}-8 x+9 y^{2}-72 y+112=0\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{8}{3-2 \cos \theta}\)

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(2 x^{2}+8 x y-1=0, \theta=30^{\circ}\)

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. \((x+1)^{2}=2(y+4)\)

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. \(4 x^{2}-24 x-36 y^{2}-360 y+864=0\)

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. \(9 x^{2}-54 x+9 y^{2}-54 y+81=0\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{3}{2+5 \quad \cos \theta}\)

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(-2 x^{2}+8 x y+1=0, \theta=45^{\circ}\)

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. \((x+4)^{2}=24(y+1)\)

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. \(-4 x^{2}+24 x+16 y^{2}-128 y+156=0\)

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. \(4 x^{2}-24 x+36 y^{2}-360 y+864=0\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{4}{2+2 \quad \sin \theta}\)

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(4 x^{2}+\sqrt{2} x y+4 y^{2}+y+2=0, \theta=45^{\circ}\)

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. \((y+4)^{2}=16(x+4)\)

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. \(-4 x^{2}+40 x+25 y^{2}-100 y+100=0\)

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. \(4 x^{2}+24 x+16 y^{2}-128 y+228=0\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{3}{8-8 \quad \cos \theta}\)

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. \(x^{2}+3 \sqrt{3} x y+4 y^{2}+y-2=0\)

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. \(y^{2}+12 x-6 y+21=0\)

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

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. \(4 x^{2}+40 x+25 y^{2}-100 y+100=0\)

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{2}{6+7 \quad \cos \theta}\)

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. \(4 x^{2}+2 \sqrt{3} x y+6 y^{2}+y-2=0\)

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. \(x^{2}-4 x-24 y+28=0\)