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

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

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

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{8}{3-2 \cos \theta} $$

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

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, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{3}{2+5 \cos \theta} $$

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{3}{2+5 \cos \theta} $$

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

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, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{4}{2+2 \sin \theta} $$

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

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{4}{2+2 \sin \theta} $$

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

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, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{3}{8-8 \cos \theta} $$

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

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{3}{8-8 \cos \theta} $$

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, 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. $$ x^{2}+2 x-100 y^{2}-1000 y+2401=0 $$

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, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{2}{6+7 \cos \theta} $$

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

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{2}{6+7 \cos \theta} $$

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. $$ x^{2}-4 x-24 y+28=0 $$

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, 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. $$ -9 x^{2}+72 x+16 y^{2}+16 y+4=0 $$

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, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{5}{5-11 \sin \theta} $$

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

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{5}{5-11 \sin \theta} $$

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, 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. $$ 4 x^{2}+24 x-25 y^{2}+200 y-464=0 $$

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, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(5+2 \cos \theta)=6 $$

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

Convert the polar equation of a conic section to a rectangular equation. $$ r(5+2 \cos \theta)=6 $$