Chapter 10

Convert the polar coordinates to rectangular coordinates. $$(-1,5 \pi / 6)$$

Find the eccentricity of the conic whose equation is given. $$\frac{(x-6)^{2}}{10}-\frac{y^{2}}{40}=1$$

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. $$4 x^{2}+3 y^{2}=12$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$17 x^{2}-48 x y+31 y^{2}+49=0$$

In Exercises \(11-16,\) find the focus and directrix of the parabola. $$y^{2}+8 x=0$$

Convert the polar coordinates to rectangular coordinates. $$(2,0)$$

Find the eccentricity of the conic whose equation is given. $$4 x^{2}+9 y^{2}-24 x+36 y+36=0$$

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. $$9 x^{2}+4 y^{2}=72$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$52 x^{2}-72 x y+73 y^{2}=200$$

In Exercises \(11-16,\) find the focus and directrix of the parabola. $$x^{2}-3 y=0$$

Convert the polar coordinates to rectangular coordinates. $$(1.5,5)$$

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

Find the eccentricity of the conic whose equation is given. $$16 x^{2}-9 y^{2}-32 x+36 y+124=0$$

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. $$\frac{y^{2}}{49}+\frac{x^{2}}{81}=1$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$9 x^{2}+24 x y+16 y^{2}+90 x-130 y=0$$

Find the eccentricity of the conic whose equation is given. $$4 x^{2}-5 y^{2}-16 x-50 y+71=0$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$x^{2}+10 x y+y^{2}+1=0$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$y-3=x^{2}$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=t-3, \quad y=2 t+1, \quad t \geq 0$$

Convert the polar coordinates to rectangular coordinates. $$(-4,-\pi / 7)$$

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. $$4 x^{2}+4 y^{2}=1$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$23 x^{2}+26 \sqrt{3} x y-3 y^{2}-16 x+16 \sqrt{3} y+128=0$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$x+(y+1)^{2}=2$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=t+5, \quad y=\sqrt{t}, \quad t \geq 0$$

(a) Graph these hyperbolas (on the same screen if possible): $$ \frac{y^{2}}{4}-\frac{x^{2}}{1}=1, \quad \frac{y^{2}}{4}-\frac{x^{2}}{12}=1, \quad \frac{y^{2}}{4}-\frac{x^{2}}{96}=1 $$ (b) Compute the eccentricity of each hyperbola in part (a). (c) On the basis of parts (a) and (b), how is the shape of a hyperbola related to its eccentricity?

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. $$x^{2}+4 y^{2}=1$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$x^{2}+2 x y+y^{2}+12 \sqrt{2} x-12 \sqrt{2} y=0$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=-2+t^{2}, \quad y=1+2 t^{2}, \quad \text { any real number } t$$

Convert the rectangular coordinates to polar coordinates. $$(3 \sqrt{3},-3)$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) foci on \(x\) -axis; \(x\) -intercepts \(\pm 7 ; y\) -intercepts \(\pm 2.\)

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$17 x^{2}-12 x y+8 y^{2}-80=0$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$3 x-2=(y+3)^{2}$$

Find the equation of the hyperbola that satisfies the given conditions. Center (0,0)\(; x\) -intercepts \(\pm 3 ;\) asymptote \(y=2 x\)

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=t^{2}+1, \quad y=t^{2}-1, \quad \text { any real number } t$$

Convert the rectangular coordinates to polar coordinates. $$(2 \sqrt{3}, 2)$$

Sketch the graph of the equation and label the vertices. $$r=\frac{5}{3+2 \sin \theta}$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) foci on \(y\) -axis; \(x\) -intercepts \(\pm 1 ; y\) -intercepts \(\pm 8.\)

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$11 x^{2}-24 x y+4 y^{2}+30 x+40 y-45=0$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$2 x-1=-6(y+1)^{2}$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=t^{3}-3 t^{2}+2 t, \quad y=t-1, \quad \text { any real number } t$$

Convert the rectangular coordinates to polar coordinates. $$(1,1)$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) foci on \(x\) -axis; major axis of length \(12 ;\) minor axis of length \(8 .\)

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$3 x^{2}+2 \sqrt{3} x y+y^{2}+4 x-4 \sqrt{3} y-16=0$$

Find the equation of the hyperbola that satisfies the given conditions. Center (0,0)\(;\) vertex (2,0)\(;\) passing through \((4, \sqrt{3})\)

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=8 t^{3}-4 t^{2}+3, \quad y=2 t-4, \quad \text { any real number } t$$

Convert the rectangular coordinates to polar coordinates. $$(\sqrt{2},-\sqrt{2})$$

Sketch the graph of the equation and label the vertices. $$r=\frac{5}{1+\cos \theta}$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) foci on \(y\) -axis; major axis of length \(20 ;\) minor axis of length 18.

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$3 x^{2}+2 \sqrt{2} x y+2 y^{2}-12=0$$

Find the equation of the hyperbola that satisfies the given conditions. Center (0,0)\(;\) vertex \((0, \sqrt{12}) ;\) passing through \((2 \sqrt{3}, 6)\)