91Ó°ÊÓ

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$9 x^{2}+4 y^{2}=1$$

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

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\sin ^{2} t, \quad y=\cos t$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$4 y^{2}-x^{2}=1$$

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

(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. $$x^{2}+2 x y+y^{2}+x-y=0$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x^{2}+6 y=0$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\cos t, \quad y=\cos 2 t$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$\frac{1}{2} x^{2}+\frac{1}{8} y^{2}=\frac{1}{4}$$

(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. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}=16$$