91Ó°ÊÓ

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

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

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

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

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

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

(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. $$ x^{2}-y^{2}=10(x-y)+1 $$

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: \(x=2\)