Chapter 7

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=e^{t}, y=e^{-t}, \text { for } t \text { in }(-\infty, \infty)$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$x^{2}=25+y^{2}$$

Find an equation for each ellipse. \(x\) -intercepts \((\pm 3,0) ; y\) -intercepts \((0, \pm 6)\)

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=e^{2 t}, y=e^{t}, \text { for } t \text { in }(-\infty, \infty)$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$9 x^{2}+36 y^{2}=36$$

Find an equation for each ellipse. Endpoints of major axis at ( \(6,0\) ) and ( \(-6,0\) ); \(c=4\)

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=\frac{1}{\sqrt{t+2}}, y=\frac{t}{t+2}, \text { for } t \text { in }(-2, \infty)$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$\frac{x^{2}}{4}+\frac{y^{2}}{4}=-1$$

Graph each circle by hand if possible. Give the domain and range. $$x^{2}+y^{2}=4$$

Find an equation for each ellipse. Vertices \((0,5)\) and \((0,-5) ; b=2\)

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=\frac{t}{t-1}, y=\frac{1}{\sqrt{t-1}}, \text { for } t \text { in }(1, \infty)$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$\frac{(x-4)^{2}}{8}+\frac{(y+1)^{2}}{2}=0$$

Graph each circle by hand if possible. Give the domain and range. $$x^{2}+y^{2}=36$$

Find an equation for each ellipse. Center \((3,-2) ; a=5 ; c=3 ;\) major axis vertical

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t+2, y=\frac{1}{t+2}, \text { for } t \neq-2$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$y^{2}-4 y=x+4$$

Graph each circle by hand if possible. Give the domain and range. $$x^{2}+y^{2}=0$$

Find an equation for each ellipse. Center \((2,0)\); minor axis of length 6 ; major axis horizontal and of length 9

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t-3, y=\frac{2}{t-3}, \text { for } t \neq 3$$

Find an equation for each ellipse. Major axis of length 6 ; foci \((0,2)\) and \((0,-2)\)

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t^{2}, y=2 \ln t, \text { for } t \text { in }(0, \infty)$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$3 x^{2}+6 x+3 y^{2}-12 y=12$$

Graph each circle by hand if possible. Give the domain and range. $$(x-2)^{2}+y^{2}=36$$

Find an equation for each ellipse. Minor axis of length \(4 ;\) foci \((-5,0)\) and \((5,0)\)

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=\ln t, y=3 \ln t, \text { for } t \text { in }(0, \infty)$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$-4 x^{2}+8 x+y^{2}+6 y=-6$$

Graph each circle by hand if possible. Give the domain and range. $$(x+2)^{2}+(y-5)^{2}=16$$

Find an equation for each ellipse. Center \((5,2) ;\) minor axis vertical, with length \(8 ; c=3\)

Graph each circle by hand if possible. Give the domain and range. $$(x-5)^{2}+(y+4)^{2}=49$$

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$y=2 x+3$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$4 x^{2}-8 x+9 y^{2}-36 y=-4$$

Find an equation for each ellipse. Center \((-3,6) ;\) major axis vertical, with length \(10 ; c=2\)

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$y=\frac{3}{2} x-4$$

Graph each circle by hand if possible. Give the domain and range. $$(x-4)^{2}+(y-3)^{2}=25$$

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$3 x^{2}+12 x+3 y^{2}=0$$

Find an equation for each ellipse. Vertices ( \(4,9\) ) and (4, 1); minor axis of length 6

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$y=\sqrt{3 x+2}, x \text { in }\left[-\frac{2}{3}, \infty\right)$$

Graph each circle by hand if possible. Give the domain and range. $$(x+3)^{2}+(y+2)^{2}=36$$

Identify the type of conic section consisting of the set of all points in the plane for which the sum of the distances from the points \((5,0)\) and \((-5,0)\) is 14

Find an equation for each ellipse. Fociat \((-3,-3)\) and \((7,-3)\); the point ( \(2,1\) ) on ellipse

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$y=(x+1)^{2}+1$$

Graph each circle by hand if possible. Give the domain and range. $$(x-1)^{2}+(y+2)^{2}=16$$

Identify the type of conic section consisting of the set of all points in the plane for which the absolute value of the difference of the distances from the points \((3,0)\) and \((-3,0)\) is 2

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$9 x^{2}+18 x+4 y^{2}-8 y-23=0$$

Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point \((3,0)\) is one and one-half times the distance from the line \(x=\frac{4}{3}\)

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$x=y^{3}+1$$

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$9 x^{2}-36 x+16 y^{2}-64 y-44=0$$

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$x=2(y-3)^{2}-4$$

Identify the type of conic section consisting of the set of all points in the plane for which the distance from the point \((2,0)\) is one-third the distance from the line \(x=10\)

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$4 x^{2}+8 x+y^{2}+2 y+1=0$$