Chapter 8

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. \(r=\frac{3}{4-4 \cos \theta}\)

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation. \(y=\frac{x^{2}}{2}, \theta=30^{\circ}\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(-5(x+5)^{2}=4(y+5)\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(\frac{(x-2)^{2}}{8}-\frac{(y+3)^{2}}{27}=1\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(\frac{(x-2)^{2}}{64}+\frac{(y-4)^{2}}{16}=1\)

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. \(r=\frac{2}{1-\sin \theta}\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(-6(y+5)^{2}=4(x-4)\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(\frac{(y-3)^{2}}{9}-\frac{(x-3)^{2}}{9}=1\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(\frac{(x+3)^{2}}{9}+\frac{(y-3)^{2}}{9}=1\)

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. \(r=\frac{6}{3+2 \quad \sin \theta}\)

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation. \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1, \theta=30^{\circ}\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(y^{2}-6 y-8 x+1=0\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(-4 x^{2}-8 x+16 y^{2}-32 y-52=0\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(\frac{x^{2}}{2}+\frac{(y+1)^{2}}{5}=1\)

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. \(r(1+\cos \theta)=5\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(x^{2}+8 x+4 y+20=0\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(4 x^{2}-8 x+16 y^{2}-32 y-44=0\)

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. \(r(3-4 \sin \theta)=9\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(3 x^{2}+30 x-4 y+95=0\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(-x^{2}+8 x+4 y^{2}-40 y+88=0\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(x^{2}-8 x+25 y^{2}-100 y+91=0\)

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. \(r(3-2 \sin \theta)=6\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(y^{2}-8 x+10 y+9=0\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(64 x^{2}+128 x-9 y^{2}-72 y-656=0\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(x^{2}+8 x+4 y^{2}-40 y+112=0\)

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. \(r(6-4 \cos \theta)=5\)

For the following exercises, graph the equation relative to the \(x^{\prime} y^{\prime}\) system in which the equation has no \(x^{\prime} y^{\prime}\) term. \(4 x^{2}-3 \sqrt{3} x y+y^{2}-22=0\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(x^{2}+4 x+2 y+2=0\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(16 x^{2}+64 x-4 y^{2}-8 y-4=0\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(64 x^{2}+128 x+9 y^{2}-72 y-368=0\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=4 ; \quad e=\frac{1}{5}\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(y^{2}+2 y-12 x+61=0\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(-100 x^{2}+1000 x+y^{2}-10 y-2575=0\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(16 x^{2}+64 x+4 y^{2}-8 y+4=0\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-4 ; \quad e=5\)

For the following exercises, graph the parabola, labeling the focus and the directrix. \(-2 x^{2}+8 x-4 y-24=0\)

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. \(4 x^{2}+16 x-4 y^{2}+16 y+16=0\)

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(100 x^{2}+1000 x+y^{2}-10 y+2425=0\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=2 ; \quad e=2\)

For the following exercises, graph the equation relative to the \(x^{\prime} y^{\prime}\) system in which the equation has no \(x^{\prime} y^{\prime}\) term. \(21 x^{2}+2 \sqrt{3} x y+19 y^{2}-18=0\)

For the following exercises, given information about the graph of the hyperbola, find its equation. Vertices at (3,0) and (-3,0) and one focus at (5,0) .

For the following exercises, graph the given ellipses, noting center, vertices, and foci. \(4 x^{2}+16 x+4 y^{2}+16 y+16=0\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=-2 ; \quad e=\frac{1}{2}\)

For the following exercises, graph the equation relative to the \(x^{\prime} y^{\prime}\) system in which the equation has no \(x^{\prime} y^{\prime}\) term. \(16 x^{2}+24 x y+9 y^{2}-130 x+90 y=0\)

For the following exercises, find the equation of the parabola given information about its graph. Vertex is (0,0)\(;\) directrix is \(x=4,\) focus is (-4,0)

For the following exercises, given information about the graph of the hyperbola, find its equation. Vertices at (0,6) and (0,-6) and one focus at (0,-8) .

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center at the origin, symmetric with respect to the \(x\) - and \(y\) -axes, focus at (4,0) , and point on graph (0,3) .

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=1 ; \quad e=1\)

For the following exercises, find the equation of the parabola given information about its graph. Vertex is (2,2)\(;\) directrix is \(x=2-\sqrt{2},\) focus is \((2+\sqrt{2}, 2)\)

For the following exercises, given information about the graph of the hyperbola, find its equation. Vertices at (1,1) and (11,1) and one focus at (12,1) .