Chapter 8

Explain how eccentricity determines which conic section is given.

What effect does the \(x y\) term have on the graph of a conic section?

Define a parabola in terms of its focus and directrix.

Define a hyperbola in terms of its foci.

Define an ellipse in terms of its foci.

If a conic section is written as a polar equation, what must be true of the denominator?

If the equation of a conic section is written in the form \(A x^{2}+B y^{2}+C x+D y+E=0\) and \(A B=0\), what can we conclude?

If the equation of a parabola is written in standard form and \(p\) is positive and the directrix is a vertical line, then what can we conclude about its graph?

What can we conclude about a hyperbola if its asymptotes intersect at the origin?

If the equation of a conic section is written in the form \(A x^{2}+B x y+C y^{2}+D x+E y+F=0,\) and \(B^{2}-4 A C>0,\) what can we conclude?

If the equation of a parabola is written in standard form and \(p\) is negative and the directrix is a horizontal line, then what can we conclude about its graph?

What must be true of the foci of a hyperbola?

What special case of the ellipse do we have when the major and minor axis are of the same length?

If the transverse axis of a hyperbola is vertical, what do we know about the graph?

As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?

Where must the center of hyperbola be relative to its foci?

What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the \(y\) -axis?

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{6}{1-2 \cos \theta}\)

For the following exercises, determine which conic section is represented based on the given equation. \(9 x^{2}+4 y^{2}+72 x+36 y-500=0\)

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(y^{2}=4-x^{2}\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{3}{4-4 \quad \sin \theta}\)

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

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(y=4 x^{2}\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{8}{4-3 \quad \cos \theta}\)

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

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(3 x^{2}-6 y^{2}=12\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{5}{1+2 \sin \theta}\)

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

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \((y-3)^{2}=8(x-2)\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{16}{4+3 \quad \cos \theta}\)

For the following exercises, determine which conic section is represented based on the given equation. \(4 y^{2}-5 x+9 y+1=0\)

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(y^{2}+12 x-6 y-51=0\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{3}{10+10 \cos \theta}\)

For the following exercises, determine which conic section is represented based on the given equation. \(2 x^{2}+3 y^{2}-8 x-12 y+2=0\)

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. \(x=8 y^{2}\)

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. \(\frac{x^{2}}{25}-\frac{y^{2}}{36}=1\)

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

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

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

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

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

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

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

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

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