91Ó°ÊÓ

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

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

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 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. $$x^{2}=y$$

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

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

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

1–8 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 \((2,0)\)