91Ó°ÊÓ

\(35-38\) Use a graphing device to graph the conic. $$ 2 x^{2}-4 x+y+5=0 $$

Find an equation for the hyperbola that satisfies the given conditions. Vertices: \(( \pm 1,0),\) asymptotes: \(y=\pm 5 x\)

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(y=-10\)

Find an equation for the ellipse that satisfies the given conditions. Length of major axis: 4 , length of minor axis: \(2,\) foci on \(y\) -axis

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{7}{2-5 \sin \theta} $$

Solve the equations $$ \begin{aligned} x &=X \cos \phi-Y \sin \phi \\ y &=X \sin \phi+Y \cos \phi \end{aligned} $$ for \(X\) and \(Y\) in terms of \(x\) and \(y\) . [Hint: To begin, multiply the first equation by \(\cos \phi\) and the second by \(\sin \phi,\) and then add the two equations to solve for \(X . ]\)

Find an equation for the hyperbola that satisfies the given conditions. Vertices: \((0, \pm 6),\) asymptotes: \(y=\pm \frac{1}{3} X\)

Show that the graph of the equation $$ \sqrt{x}+\sqrt{y}=1 $$ is part of a parabola by rotating the axes through an angle of \(45^{\circ} .[\text { Hint: First convert the equation to one that does not }\) involve radicals. \(]\)

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{8}{3+\cos \theta} $$

Find an equation for the ellipse that satisfies the given conditions. Length of major axis: \(6,\) length of minor axis: \(4,\) foci on \(x\) -axis