Chapter 11

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$y^{2}-16 x^{2}=1$$

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ y^{2}+14 y+4 x+45=0 $$

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ \frac{(x+2)^{2}}{25}+\frac{(y-3)^{2}}{4}=1 $$

Exer. 9-12: Change the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta \leq 2 \pi\). (a) \((7,-7 \sqrt{3})\) (b) \((5,5)\)

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$\frac{(y+2)^{2}}{9}-\frac{(x+2)^{2}}{4}=1$$

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ x^{2}+20 y=10 $$

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ 4 x^{2}+9 y^{2}-32 x-36 y+64=0 $$

Exer. 9-12: Change the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta \leq 2 \pi\). (a) \((-2 \sqrt{2},-2 \sqrt{2})\) (b) \((-4,4 \sqrt{3})\)

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$\frac{(x-3)^{2}}{25}-\frac{(y-1)^{2}}{4}=1$$

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ y^{2}-4 y-2 x-4=0 $$

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ x^{2}+2 y^{2}+2 x-20 y+43=0 $$

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x=-3 $$

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$144 x^{2}-25 y^{2}+864 x-100 y-2404=0$$

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ 25 x^{2}+4 y^{2}-250 x-16 y+541=0 $$

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ y=2 $$

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$y^{2}-4 x^{2}-12 y-16 x+16=0$$

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ 4 x^{2}+y^{2}=2 y $$

\(x=\sin t, \quad y=\csc t ; \quad 0

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x^{2}+y^{2}=16 $$

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$4 y^{2}-x^{2}+40 y-4 x+60=0$$

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x^{2}+y^{2}=2 $$

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$25 x^{2}-9 y^{2}+100 x-54 y+10=0$$

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ y^{2}=6 x $$

\(x=-2 \sqrt{1-t^{2}}, y=t ; \quad|t| \leq 1\)

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x^{2}=8 y $$

\(x=t, \quad y=\sqrt{t^{2}-2 t+1} ; \quad 0 \leq t \leq 4\)

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x+y=3 $$

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Focus \(F(2,0)\), $$ \text { directrix } x=-2 $$

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(\pm 8,0)\), $$ \text { foci } F(\pm 5,0) $$

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ 2 y=-x+4 $$

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Focus \(F(0,-4)\) directrix \(y=4\)

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 7)\), foci \(F(0, \pm 2)\)

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ 2 y=-x $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 4)\) vertices \(V(0, \pm 1)\)

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Focus \(F(6,4)\) $$ \text { directrix } y=-2 $$

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 5)\), minor axis of length 3

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ y=6 x $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 8,0)\) vertices \(V(\pm 5,0)\)

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. $$ \text { Focus } F(-3,-2), \quad \text { directrix } y=1 $$

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 3,0)\), minor axis of length 2

\(x=e^{t}, \quad y=e^{-2 t}, \quad t\) in \(\mathbb{R}\)

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ y^{2}-x^{2}=4 $$

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

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. $$ \text { Vertex } V(3,-5), \quad \text { directrix } x=2 $$

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 6)\), passing through \((3,2)\)

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x y=8 $$

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

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex \(V(-2,3)\) directrix \(y=5\)

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=\frac{1}{3}, \quad r=2 \sec \theta $$

(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).