91Ó°ÊÓ

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 it 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. $$x^{2}+4 y^{2}+20 x-40 y+300=0$$

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. $$r=\frac{5}{2-3 \sin \theta}$$

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

Solve the equations $$\begin{array}{l} x=X \cos \phi-Y \sin \phi \\ y=X \sin \phi+Y \cos \phi \end{array}$$ for \(X\) and \(Y\) in terms of \(x\) and \(y .\) [Hint: To begin, multiply the first equation by \(\cos \phi\) and the second by \(\sin \phi,\) and then add the two equations to solve for \(X .]\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{7}{2-5 \sin \theta}$$

Use a graphing device to graph the conic. $$2 x^{2}-4 x+y+5=0$$

Find an equation for the hyperbola that satisfies the given conditions. Vertices: \((\pm 1,0),\) asymptotes: \(y=\pm 5 x\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(y=-10\)

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