91Ó°ÊÓ

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=\frac{3}{4}, \quad r \sin \theta=5$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{2}{3}, \quad\) vertices on the \(y\) -axis, passing through \((1,4)\)

(a) Describe the graph of a curve \(C\) that has the parametrization $$x=-2+3 \sin t, \quad y=3-3 \cos t ; \quad 0 \leq t \leq 2 \pi$$ (b) Change the parametrization to $$x=-2-3 \sin t, \quad y=3+3 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a). (c) Change the parametrization to $$x=-2+3 \sin t, \quad y=3+3 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a).

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$(x+2)^{2}+y^{2}=4$$

Find an equation of the parabola that satisfies the given conditions. $$\text { Vertex } V(4,7), \quad \text { focus } F(4,2)$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm 5, \quad\) asymptotes \(y=\pm 2 x\)

Find a polar equation of the parabola with focus at the pole and the given vertex. $$V\left(4, \frac{\pi}{2}\right)$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm 2\) \(y\) -intercepts \(\pm \frac{1}{3}\)

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}+(y+3)^{2}=9$$

Find an equation of the parabola that satisfies the given conditions. Vertex at the origin, symmetric with respect to the \(y\) -axis, and passing through the point \((2,-3)\)