Chapter 7

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$\frac{x^{2}}{4}=1+\frac{y^{2}}{9}$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$12 x^{2}+8 y^{2}=96$$

Find the center-radius form for each circle satisfying the given conditions. $$\text { Center }\left(-\frac{1}{2},-\frac{1}{4}\right) ; \text { radius } \frac{12}{5}$$

For each plane curve, use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then, find a rectangular equation for the curve. \(x=t+2, y=-\frac{1}{2} \sqrt{9-t^{2}},\) for \(t\) in \([-3,3]\) window: \([-6,6]\) by \([-4,4]\)

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$\frac{(x+3)^{2}}{16}+\frac{(y-2)^{2}}{16}=1$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{25 y^{2}}{36}+\frac{64 x^{2}}{9}=1$$

Find the center-radius form for each circle satisfying the given conditions. Center \((-1,2) ;\) passing through \((2,6)\)

For each plane curve, use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then, find a rectangular equation for the curve. \(x=t, y=\sqrt{4-t^{2}},\) for \(t\) in \([-2,2]\) window: \([-6,6]\) by \([-4,4]\)

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}=25-y^{2}$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{16 y^{2}}{9}+\frac{121 x^{2}}{25}=1$$

Find the center-radius form for each circle satisfying the given conditions. Center \((2,-7) ;\) passing through \((-2,-4)\)

For each plane curve, use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then, find a rectangular equation for the curve. \(x=t, y=\frac{1}{t},\) for \(t\) in \((-\infty, 0) \cup(0, \infty)\) window: \([-6,6]\) by \([-4,4]\)

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{(x-1)^{2}}{9}+\frac{(y+3)^{2}}{25}=1$$

Find the center-radius form for each circle satisfying the given conditions. Center \((-3,-2) ;\) tangent to the \(x\) -axis (Hint: "tangent to" means touching at one point.)

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}-6 x+y=0$$

For each plane curve, use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then, find a rectangular equation for the curve. \(x=2 t-1, y=\frac{1}{t},\) for \(t\) in \((-\infty, 0) \cup(0, \infty)\) window: \([-6,6]\) by \([-4,4]\)

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$11-3 x=2 y^{2}-8 y$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{(x+3)^{2}}{16}+\frac{(y-2)^{2}}{36}=1$$

Find the center-radius form for each circle satisfying the given conditions. Center \((5,-1) ;\) tangent to the \(y\) -axis

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t, y=t-1, \text { for } t \text { in }(-\infty, \infty)$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{(x-2)^{2}}{16}+\frac{(y-1)^{2}}{9}=1$$

Find the center-radius form for each circle satisfying the given conditions. Describe the graph of the following equation. $$ (x-3)^{2}+(y-3)^{2}=0 $$

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$4(x-3)^{2}+3(y+4)^{2}=0$$

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

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$2 x^{2}-8 x+2 y^{2}+20 y=12$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{(x+3)^{2}}{25}+\frac{(y+2)^{2}}{36}=1$$

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t^{2}, y=t+1, \text { for } t \text { in }(-\infty, \infty)$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{(x+1)^{2}}{64}+\frac{(y-2)^{2}}{49}=1$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-1,3) \text { and }(5,-9)$$

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

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}+2 x=-4 y$$

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{(x-4)^{2}}{9}+\frac{(y+2)^{2}}{4}=1$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-4,5) \text { and }(6,-9)$$

Find an equation for each ellipse. \(x\) -intercepts \((\pm 4,0) ;\) foci \((\pm 2,0)\)

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t^{2}, y=4 t^{3}, \text { for } t \text { in }(-\infty, \infty)$$

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$6 x^{2}-12 x+6 y^{2}-18 y+25=0$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-5,-7) \text { and }(1,1)$$

Find an equation for each ellipse. \(y\) -intercepts \((0, \pm 3) ;\) foci \((0, \pm \sqrt{3})\)

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

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$4 x^{2}-24 x+5 y^{2}+10 y+41=0$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-3,-2) \text { and }(1,-4)$$

Find an equation for each ellipse. \(y\) -intercepts \((0, \pm 2 \sqrt{2}) ;\) foci \((0, \pm 2)\)

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

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

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-5,0) \text { and }(5,0)$$

Find an equation for each ellipse. \(x\) -intercepts \((\pm 3 \sqrt{2}, 0) ;\) foci \((\pm 2 \sqrt{3}, 0)\)

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=\sqrt{t}, y=t^{2}-1, \text { for } t \text { in }[0, \infty)$$

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

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. \((0,9)\) and \((0,-9)\)

Find an equation for each ellipse. \(x\) -intercepts \((\pm 4,0) ; y\) -intercepts \((0, \pm 2)\)