91Ó°ÊÓ

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(0,2), \quad \phi=55^{\circ}$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{x^{2}}{4}-\frac{y^{2}}{16}=1$$

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(\sqrt{2}, 4 \sqrt{2}), \quad \phi=45^{\circ}$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$$

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. $$x=t^{2}, \quad y=t^{4}+1$$

Find the vertex, focus, and directrix of the parabola, and sketch the graph. $$(y+5)^{2}=-6 x+12$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.6,\) directrix \(r=2 \csc \theta\)

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$y^{2}-\frac{x^{2}}{25}=1$$

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