Chapter 10

Find a viewing window that shows a complete graph of the curve. $$x=t^{2}-4, \quad y=t / 2, \quad-2 \leq t \leq 3$$

Assume that the graph of the equation is a nondegenerate conic section. Without graphing, determine whether the graph an ellipse, hyperbola, or parabola. $$x^{2}-2 x y+3 y^{2}-1=0$$

Plot the point whose polar coordinates are given. $$(1, \pi / 4)$$

Plot the point whose polar coordinates are given. $$(2,-3 \pi / 4)$$

Find a viewing window that shows a complete graph of the curve. $$x=2 t, \quad y=t^{2}-1, \quad-1 \leq t \leq 2$$

Assume that the graph of the equation is a nondegenerate conic section. Without graphing, determine whether the graph an ellipse, hyperbola, or parabola. $$x^{2}+2 x y+y^{2}+2 \sqrt{2} x-2 \sqrt{2} y=0$$

Plot the point whose polar coordinates are given. $$(-2,2 \pi / 3)$$

Assume that the graph of the equation is a nondegenerate conic section. Without graphing, determine whether the graph an ellipse, hyperbola, or parabola. $$2 x^{2}-4 x y+5 y^{2}-6=0$$

Plot the point whose polar coordinates are given. $$(-3,-5 \pi / 3)$$

Assume that the graph of the equation is a nondegenerate conic section. Without graphing, determine whether the graph an ellipse, hyperbola, or parabola. $$17 x^{2}-48 x y+31 y^{2}+50=0$$

Plot the point whose polar coordinates are given. $$(3, \pi / 6)$$

Find a viewing window that shows a complete graph of the curve. $$x=\sin 4 t, \quad y=\sin 3 t, \quad 0 \leq t \leq 2 \pi$$

Assume that the graph of the equation is a nondegenerate conic section. Without graphing, determine whether the graph an ellipse, hyperbola, or parabola. $$2 x^{2}-4 x y-2 y^{2}+3 x+5 y-10=0$$

Identify the conic section whose equation is given\(;\) if it is an ellipse or hyperbola, state its eccentricity. $$r=\frac{12}{3+4 \sin \theta}$$

Find a viewing window that shows a complete graph of the curve. $$x=\cos 3 t+t, \quad y=\cos 4 t+t, \quad-\pi \leq t \leq \pi$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$9 x^{2}+4 y^{2}+54 x-8 y+49=0$$

Find the center and radius of the circle whose equation is given. $$x^{2}+y^{2}+8 x-6 y-15=0$$

In Exercises \(7-10,\) find the equation of the parabola. focus (4,0)\(;\) directrix \(x=-4\)

Identify the conic section whose equation is given\(;\) if it is an ellipse or hyperbola, state its eccentricity. $$r=\frac{-10}{2+3 \cos \theta}$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$4 x^{2}+5 y^{2}-8 x+30 y+29=0$$

Find the center and radius of the circle whose equation is given. $$15 x^{2}+15 y^{2}=10$$

In Exercises \(7-10,\) find the equation of the parabola. focus (-5,0)\(;\) directrix \(x=5\)

List four other pairs of polar coordinates for the given point, each with a different combination of signs (that is, \(r > 0, \theta > 0 ; r > 0, \theta < 0 ; r < 0, \theta > 0 ; r < 0, \theta < 0)\). $$(2,-2 \pi / 3)$$

Identify the conic section whose equation is given\(;\) if it is an ellipse or hyperbola, state its eccentricity. $$r=\frac{8}{3+3 \sin \theta}$$

Find a viewing window that shows a complete graph of the curve. $$x=4 \sin 2 t+9, \quad y=6 \cos t-8, \quad 0 \leq t \leq 2 \pi$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$4 y^{2}-x^{2}+6 x-24 y+11=0$$

Find the center and radius of the circle whose equation is given. $$x^{2}+y^{2}+6 x-4 y-15=0$$

Find a viewing window that shows a complete graph of the curve. $$x=t^{3}-3 t-8, \quad y=3 t^{2}-15, \quad-4 \leq t \leq 4$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$x^{2}-16 y^{2}=0$$

Find the center and radius of the circle whose equation is given. $$x^{2}+y^{2}+10 x-75=0$$

Identify the conic section whose equation is given and find its graph. List its vertices, foci, and asymptotes. $$4 x^{2}-y^{2}=16$$

List four other pairs of polar coordinates for the given point, each with a different combination of signs (that is, \(r > 0, \theta > 0 ; r > 0, \theta < 0 ; r < 0, \theta > 0 ; r < 0, \theta < 0)\). $$(\sqrt{3}, 3 \pi / 4)$$

Identify the conic section whose equation is given\(;\) if it is an ellipse or hyperbola, state its eccentricity. $$r=\frac{2}{6-4 \cos \theta}$$

Find a viewing window that shows a complete graph of the curve. $$\begin{array}{ll} x=6 \cos t+12 \cos ^{2} t, & y=8 \sin t+8 \sin t \cos t \\ 0 \leq t \leq 2 \pi \end{array}$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$3 y^{2}-x-2 y+1=0$$

Find the center and radius of the circle whose equation is given. $$x^{2}+y^{2}+25 x+10 y=-12$$

List four other pairs of polar coordinates for the given point, each with a different combination of signs (that is, \(r > 0, \theta > 0 ; r > 0, \theta < 0 ; r < 0, \theta > 0 ; r < 0, \theta < 0)\). $$(-3,7 \pi / 6)$$

Find a viewing window that shows a complete graph of the curve.\(x=12 \cos t, \quad y=12 \sin 2 t, \quad 0 \leq t \leq 2 \pi$$x=12 \cos t, \quad y=12 \sin 2 t, \quad 0 \leq t \leq 2 \pi$$x=12 \cos t, \quad y=12 \sin 2 t, \quad 0 \leq t \leq 2 \pi\)

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$x^{2}-6 x+y+5=0$$

In Exercises \(11-16,\) find the focus and directrix of the parabola. $$x=.5 y^{2}$$

Convert the polar coordinates to rectangular coordinates. $$(3, \pi / 3)$$

Find a viewing window that shows a complete graph of the curve. $$\begin{array}{ll} x=6 \cos t+5 \cos 3 t, & y=6 \sin t-5 \sin 3 t \\ 0 \leq t \leq 2 \pi \end{array}$$

Find the eccentricity of the conic whose equation is given. $$\frac{x^{2}}{100}+\frac{y^{2}}{99}=1$$

Identify the conic section whose equation is given and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$41 x^{2}-24 x y+34 y^{2}-25=0$$

In Exercises \(11-16,\) find the focus and directrix of the parabola. $$y=25 x^{2}$$

Find the eccentricity of the conic whose equation is given. $$\frac{(x-4)^{2}}{18}+\frac{(y+5)^{2}}{25}=1$$

Identify the conic section whose equation is given and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$\frac{x^{2}}{6}+\frac{y^{2}}{16}=1$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$x^{2}+2 \sqrt{3} x y+3 y^{2}+8 \sqrt{3} x-8 y+32=0$$

In Exercises \(11-16,\) find the focus and directrix of the parabola. $$x=-6 y^{2}$$