Chapter 12

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+6 x+12 y+9=0 $$

\(15-28=(a)\) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$ 9 x^{2}-24 x y+16 y^{2}=100(x-y-1) $$

$$ \text { Use a graphing device to graph the parabola. } $$ $$ 8 y^{2}=x $$

\(23-28\) Use a graphing device to graph the parabola. $$ 8 y^{2}=x $$

Use a graphing device to graph the hyperbola. $$ x^{2}-2 y^{2}=8 $$

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices anddirectrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the asymptotes. $$ r=\frac{20}{2-3 \sin \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 4 x^{2}+25 y^{2}-24 x+250 y+561=0 $$

\(15-28=(a)\) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$ 52 x^{2}+72 x y+73 y^{2}=40 x-30 y+75 $$

\(23-28\) Use a graphing device to graph the parabola. $$ 4 x+y^{2}=0 $$

Use a graphing device to graph the hyperbola. $$ 3 y^{2}-4 x^{2}=24 $$

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices anddirectrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the asymptotes. $$ r=\frac{6}{2+7 \cos \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 2 x^{2}+y^{2}=2 y+1 $$

\(15-28=(a)\) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$ (7 x+24 y)^{2}=600 x-175 y+25 $$

\(23-28\) Use a graphing device to graph the parabola. $$ x-2 y^{2}=0 $$

Use a graphing device to graph the hyperbola. $$ \frac{y^{2}}{2}-\frac{x^{2}}{6}=1 $$

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{4}{1+3 \cos \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 16 x^{2}-9 y^{2}-96 x+288=0 $$

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ 2 x^{2}-4 x y+2 y^{2}-5 x-5=0 $$

Use a graphing device to graph the ellipse. $$ \frac{x^{2}}{25}+\frac{y^{2}}{20}=1 $$

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F(0,2)\)

Use a graphing device to graph the hyperbola. $$ \frac{x^{2}}{100}-\frac{y^{2}}{64}=1 $$

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{8}{3+3 \cos \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 4 x^{2}-4 x-8 y+9=0 $$

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ x^{2}-2 x y+3 y^{2}=8 $$

Use a graphing device to graph the ellipse. $$ x^{2}+\frac{y^{2}}{12}=1 $$

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F\left(0,-\frac{1}{2}\right)\)

Find an equation for the hyperbola that satisfies the given conditions. Foci: \(( \pm 5,0),\) vertices: \(( \pm 3,0)\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{2}{1-\cos \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+16=4\left(y^{2}+2 x\right) $$

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ 6 x^{2}+10 x y+3 y^{2}-6 y=36 $$

Use a graphing device to graph the ellipse. $$ 6 x^{2}+y^{2}=36 $$

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F(-8,0)\)

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 10),\) vertices: \((0, \pm 8)\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{10}{3-2 \sin \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}-y^{2}=10(x-y)+1 $$

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ 9 x^{2}-6 x y+y^{2}+6 x-2 y=0 $$

Use a graphing device to graph the ellipse. $$ x^{2}+2 y^{2}=8 $$

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F(5,0)\)

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 2),\) vertices: \((0, \pm 1)\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{6}{2+\sin \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 3 x^{2}+4 y^{2}-6 x-24 y+39=0 $$

(a) Use rotation of axes to show that the following equation represents a hyperbola. $$ 7 x^{2}+48 x y-7 y^{2}-200 x-150 y+600=0 $$ (b) Find the \(X Y\) - and \(x y-\) coordinates of the center, vertices, and foci. (c) Find the equations of the asymptotes in \(X Y\) - and Xy-coordinates.

Find an equation for the ellipse that satisfies the given conditions. Foci \(:( \pm 4,0),\) vertices: \(( \pm 5,0)\)

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(x=2\)

Find an equation for the hyperbola that satisfies the given conditions. Foci: \(( \pm 6,0),\) vertices: \(( \pm 2,0)\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{5}{2-3 \sin \theta} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+4 y^{2}+20 x-40 y+300=0 $$

(a) Use rotation of axes to show that the following equation represents a parabola. $$ 2 \sqrt{2}(x+y)^{2}=7 x+9 y $$ (b) Find the \(X Y\) - and \(x y-\) coordinates of the vertex and focus. (c) Find the equation of the directrix in \(X Y\) - and \(x y\) coordinates.

Find an equation for the ellipse that satisfies the given conditions. Foci: \((0, \pm 3),\) vertices: \((0, \pm 5)\)

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(y=6\)