91Ó°ÊÓ

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi $$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$y=-2 x^{2}$$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x y=x+y, \quad \phi=\pi / 4$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(x^{2}-y^{2}+4=0\)

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 5 x^{2}+6 y^{2}=30 $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(x^{2}-4 y^{2}-8=0\)

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=-8 y^{2}$$

(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. $$x y=8$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ x^{2}+4 y^{2}=1 $$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=\frac{1}{2} y^{2}$$