Chapter 10

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

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

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

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

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

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

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

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

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

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

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$r=\csc \theta(\csc \theta-\cot \theta)$$

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

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

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) \((-4,4 \sqrt{3})\)

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

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=\sec t, \quad y=\tan t, \quad-\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. $$144 x^{2}-25 y^{2}+864 x-100 y-2404=0$$

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

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

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=\csc t, \quad y=\cot t, \quad-\pi

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

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

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

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. $$x=\cos 2 t, \quad y=\sin t, \quad-\pi \leq t \leq \pi$$

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

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

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. $$x=\cos t, \quad y=\cos 2 t, \quad-\pi \leq t \leq \pi$$

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

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x=5$$

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. $$\boldsymbol{x}=t^{2}, \quad \quad y=2 \ln t, \quad t>0$$

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

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. $$x=\cos ^{3} t, \quad y=\sin ^{3} t, \quad 0 \leq t \leq 2 \pi$$

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

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=\sin t, \quad y=\csc t, \quad 0

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

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. $$\text { Vertices }V \pm 8,0), \quad \text { foci } F(\pm 5,0)$$

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. $$x=\boldsymbol{e}^{t}, \quad \boldsymbol{y}=\boldsymbol{e}^{-t}\quad t \text { in } \mathbb{R}$$

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

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. $$\text { Vertices } V(0, \pm 7), \quad \text { foci } F(0, \pm 2)$$

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. $$x=t, \quad y=\sqrt{t^{2}-1} ; \quad|t| \geq 1$$

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

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}=5 y$$

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 5), \quad\) minor axis of length 3

Find an equation of the parabola that satisfies the given conditions. Focus \(F(2,0), \quad\) directrix \(x=-2\)

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. $$x=-\sqrt{t^{2}-1}, \quad y=t, \quad|t| \geq 1$$

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