Chapter 10

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V \pm 7,0), \quad\) minor axis of length 5

Find an equation of the parabola that satisfies the given conditions. Focus \(F(-4,0), \quad\) directrix \(x=4\)

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=-2 \sqrt{1-t^{2}}, y=t, \quad|t| \leq 1$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 5,0)\) vertices \(V \pm 3,0)\)

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

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 3,0), \quad\) minor axis of length 2

Find an equation of the parabola that satisfies the given conditions. Focus \(F(6,4), \quad\) directrix \(y=-2\)

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=\frac{6}{5} \sqrt{25-t^{2}}, \quad y=t, \quad|t| \leq 5$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 3)\) vertices \(V(0, \pm 2)\)

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

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 4), \quad\) minor axis of length 4

Find an equation of the parabola that satisfies the given conditions. Focus \(F(-3,-2),\) directrix \(y=1\)

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=\frac{1}{3}, \quad r=2 \sec \theta$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 5), \quad\) conjugate axis of length 4

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$2 y=-x$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 6), \quad\) passing through \((3,2)\)

Find an equation of the parabola that satisfies the given conditions. Vertex \(V(3,-5), \quad\) directrix \(x=2\)

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=1, \quad r \cos \theta=5$$

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$\boldsymbol{x}=2 t, \quad \quad y=8 t^{3} ; \quad-1 \leq t \leq 1$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 7,0), \quad\) conjugate axis of length 8

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$y=6 x$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(M \pm 13,0), \quad\) passing through \((5,6)\)

Find an equation of the parabola that satisfies the given conditions. Vertex \(V-2,3\) ). directrix \(x=1\)

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

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=(t+1)^{3}, \quad y=(t+2)^{2} ; \quad 0 \leq t \leq 2$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices \(V \pm 4,0), \quad\) passing through \((8,2)\)

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

Find an equation of the parabola that satisfies the given conditions. Vertex \(V(-2,3), \quad\) directrix \(y=5\)

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

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=t^{3}, \quad y=t^{2};\quad t \text { in } \mathbb{R}$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 12),\) passing through \((5,13)\)

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x y=8$$

Find an equation of the parabola that satisfies the given conditions. Vertex \(V(4,2), \quad\) directrix \(y=-6\)

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=1, r \sin \theta=-2$$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices \(V(\pm 3,0), \quad\) asymptotes \(y=\pm 2 x\)

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x y=-3$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{3}{4}\) vertices \(H 0, \pm 4)\)

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

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=\tan t, \quad y=1 ; \quad-\pi / 2

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 6), \quad\) asymptotes \(y=\pm 3 x\)

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

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{4}{7}\) vertices \(V \pm 7,0)\)

Find an equation of the parabola that satisfies the given conditions. $$\text { Vertex }M-2,1), \quad \text { focus } F(2,1)$$

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=\frac{2}{5}, \quad r=4 \csc \theta$$

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

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 10), \quad\) asymptotes \(y=\pm \frac{1}{3} x\)

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$(x-1)^{2}+y^{2}=1$$

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

Find an equation of the parabola that satisfies the given conditions. $$\text { Vertex } V(1,-2), \quad \text { focus } F(1,0)$$

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