Chapter 10

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

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. $$x=t-2, \quad y=2 t+3 ; \quad 0 \leq t \leq 5$$

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. $$8 y=x^{2}$$

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

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. $$x=1-2 t, \quad y=1+t, \quad-1 \leq t \leq 4$$

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

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) \((-4, \pi / 2)\)

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. $$x^{2}=-3 y$$

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

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

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

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

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

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. $$x=t^{3}+1, \quad y=t^{3}-1 ; \quad-2 \leq t \leq 2$$

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

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

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 vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$20 x=y^{2}$$

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

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. $$x=4 t^{2}-5, \quad y=2 t+3\quad t \text { in } \mathbb{R}$$

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

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

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

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

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. $$x=1-9 t, \quad y=3 t+1\quad t \text { in } \mathbb{R}$$

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. 1-14: 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 vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$(x-3)^{2}=\frac{1}{2}(y+1)$$

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

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. 1-14: 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 vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$(y-2)^{2}=\frac{1}{4}(x-3)$$

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

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. $$x=\sqrt{t}, \quad y=3 t+4 ; \quad t \geq 0$$

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

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

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

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

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. $$x=4 \cos t+1, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi$$

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

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