91Ó°ÊÓ

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (0,2), \quad \phi=55^{\circ} $$

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (\sqrt{2}, 4 \sqrt{2}), \quad \phi=45^{\circ} $$

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

\(5-8\) . Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$ \frac{(x+2)^{2}}{4}+y^{2}=1 $$

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ x^{2}-3 y^{2}=4, \quad \phi=60^{\circ} $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 $$

\(9-12\) . Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$ (x-3)^{2}=8(y+1) $$

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

\(9-12\) . Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$ (y+5)^{2}=-6 x+12 $$

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