Chapter 8

Identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{2}{1-\cos \theta} $$

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. $$ y=\frac{1}{4} x^{2} $$

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. $$ \frac{x^{2}}{100}+\frac{y^{2}}{64}=1 $$

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. $$ \frac{x^{2}}{100}-\frac{y^{2}}{9}=1 $$

For the following exercises, determine which conic section is represented based on the given equation. $$4 x^{2}+9 x y+4 y^{2}-36 y-125=0$$

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. \(\frac{x^{2}}{100}-\frac{y^{2}}{9}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{4}{7+2 \cos \theta} $$

Identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{4}{7+2 \cos \theta} $$

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. $$ y=-4 x^{2} $$

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. $$ x^{2}+9 y^{2}=1 $$

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. $$ \frac{y^{2}}{4}-\frac{x^{2}}{81}=1 $$

For the following exercises, determine which conic section is represented based on the given equation. $$3 x^{2}+6 x y+3 y^{2}-36 y-125=0$$

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. \(\frac{y^{2}}{4}-\frac{x^{2}}{81}=1\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(1-\cos \theta)=3 $$

Identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(1-\cos \theta)=3 $$

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. $$ x=\frac{1}{8} y^{2} $$

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, 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, determine which conic section is represented based on the given equation. $$-3 x^{2}+3 \sqrt{3} x y-4 y^{2}+9=0$$

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

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

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

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

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

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

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

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

Convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{4}{1+3 \sin \theta} $$

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

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

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

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

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

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