Chapter 11

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{12}{6+2 \sin \theta} $$

Which polar coordinates represent the same point as \((3, \pi / 3) ?\) (a) \((3,7 \pi / 3)\) (b) \((3,-\pi / 3)\) (c) \((-3,4 \pi / 3)\) (d) \((3,-2 \pi / 3)\) (e) \((-3,-2 \pi / 3)\) (f) \((-3,-\pi / 3)\)

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

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

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ 8 y=x^{2} $$

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{12}{6-2 \sin \theta} $$

Which polar coordinates represent the same point as \((4,-\pi / 2) ?\) (a) \((4,5 \pi / 2)\) (b) \((4,7 \pi / 2)\) (c) \((-4,-\pi / 2)\) (d) \((4,-5 \pi / 2)\) (e) \((-4,-3 \pi / 2)\) (f) \((-4, \pi / 2)\)

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$\frac{y^{2}}{49}-\frac{x^{2}}{16}=1$$

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

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

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{12}{2-6 \cos \theta} $$

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

Exer. 3-8: Change the polar coordinates to rectangular coordinates. (a) \((3, \pi / 4)\) (b) \((-1,2 \pi / 3)\)

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

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ 2 y^{2}=-3 x $$

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ \frac{x^{2}}{15}+\frac{y^{2}}{16}=1 $$

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{12}{2+6 \cos \theta} $$

Exer. 3-8: Change the polar coordinates to rectangular coordinates. (a) \((5,5 \pi / 6)\) (b) \((-6,7 \pi / 3)\)

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$\frac{x^{2}}{49}-\frac{y^{2}}{16}=1$$

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ x^{2}=-3 y $$

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ \frac{x^{2}}{45}+\frac{y^{2}}{49}=1 $$

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{3}{2+2 \cos \theta} $$

\(x=4 t^{2}-5, \quad y=2 t+3 ; \quad t\) in \(\mathbb{R}\)

Exer. 3-8: Change the polar coordinates to rectangular coordinates. (a) \((8,-2 \pi / 3)\) (b) \((-3,5 \pi / 3)\)

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$x^{2}-\frac{y^{2}}{24}=1$$

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ (x+2)^{2}=-8(y-1) $$

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

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{3}{2-2 \sin \theta} $$

Exer. 3-8: Change the polar coordinates to rectangular coordinates. (a) \((4,-\pi / 4)\) (b) \((-2,7 \pi / 6)\)

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$y^{2}-\frac{x^{2}}{15}=1$$

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ (x-3)^{2}=\frac{1}{2}(y+1) $$

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

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{4}{\cos \theta-2} $$

Exer. 3-8: Change the polar coordinates to rectangular coordinates. $$ \left(6, \arctan \frac{3}{4}\right) $$

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

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ (y-2)^{2}=\frac{1}{4}(x-3) $$

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

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{4 \sec \theta}{2 \sec \theta-1} $$

Exer. 3-8: Change the polar coordinates to rectangular coordinates. $$ \left(10, \arccos \left(-\frac{1}{3}\right)\right) $$

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$x^{2}-2 y^{2}=8$$

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

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ 10 y^{2}+x^{2}=5 $$

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{6 \csc \theta}{2 \csc \theta+3} $$

\(x=2-3 \sin t, \quad y=-1-3 \cos t ; \quad 0 \leq t \leq 2 \pi\)

Exer. 9-12: Change the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta \leq 2 \pi\). (a) \((-1,1)\) (b) \((-2 \sqrt{3},-2)\)

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$16 x^{2}-36 y^{2}=1$$

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

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ \frac{(x-3)^{2}}{16}+\frac{(y+4)^{2}}{9}=1 $$

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{8 \csc \theta}{2 \csc \theta-5} $$

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