91Ó°ÊÓ

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. $$x=\frac{1}{t}, \quad y=t+1$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$9 x^{2}+4 y^{2}=36$$

Find the vertex, focus, and directrix of the parabola, and sketch the graph. $$-4\left(x+\frac{1}{2}\right)^{2}=y$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$y^{2}=4 x$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.4,\) vertex at \((2,0)\)

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$4 x^{2}+25 y^{2}=100$$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$y=(x-1)^{2}, \quad \phi=45^{\circ}$$

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. $$x=t+1, \quad y=\frac{t}{t+1}$$

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

Find the vertex, focus, and directrix of the parabola, and sketch the graph. $$y^{2}=16 x-8$$