Chapter 10

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$y^{2}=4(x+2 y)$$

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{5}{2-3 \sin \theta}$$

Use a graphing device to graph the parabola. $$x^{2}=-8 y$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\cos ^{2} t, \quad y=\sin ^{2} t$$

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$\sqrt{3} x^{2}+3 x y=3$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}-4 y^{2}-2 x+16 y=20$$

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{7}{2-5 \sin \theta}$$

Use a graphing device to graph the parabola. $$y^{2}=-\frac{1}{3} x$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\cos ^{3} t, \quad y=\sin ^{3} t, \quad 0 \leq t \leq 2 \pi$$

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$153 x^{2}+192 x y+97 y^{2}=225$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}+6 x+12 y+9=0$$

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{8}{3+\cos \theta}$$

Use a graphing device to graph the parabola. $$8 y^{2}=x$$

Find parametric equations for the line with the given properties. Slope \(\frac{1}{2},\) passing through \((4,-1)\)

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$2 \sqrt{3} x^{2}-6 x y+\sqrt{3} x+3 y=0$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$4 x^{2}+25 y^{2}-24 x+250 y+561=0$$

(a) Find the eccentricity and directrix of the conic \(r=1 /(4-3 \cos \theta)\) and graph the conic and its directrix. (b) If this conic is rotated about the origin through an angle \(\pi / 3,\) write the resulting equation and draw its graph.

Use a graphing device to graph the hyperbola. $$x^{2}-2 y^{2}=8$$

Use a graphing device to graph the parabola. $$4 x+y^{2}=0$$

Find parametric equations for the line with the given properties. Slope \(-2,\) passing through \((-10,-20)\)

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$9 x^{2}-24 x y+16 y^{2}=100(x-y-1)$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$2 x^{2}+y^{2}=2 y+1$$

Graph the parabola \(r=5 /(2+2 \sin \theta)\) and its directrix. Also graph the curve obtained by rotating this parabola about its focus through an angle \(\pi / 6\)

Use a graphing device to graph the hyperbola. $$3 y^{2}-4 x^{2}=24$$

Use a graphing device to graph the parabola. $$x-2 y^{2}=0$$

Find parametric equations for the line with the given properties. Passing through \((6,7)\) and \((7,8)\)

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$52 x^{2}+72 x y+73 y^{2}=40 x-30 y+75$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$16 x^{2}-9 y^{2}-96 x+288=0$$

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6,0.8\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

Use a graphing device to graph the hyperbola. $$\frac{y^{2}}{2}-\frac{x^{2}}{6}=1$$

Use a graphing device to graph the ellipse. $$\frac{x^{2}}{25}+\frac{y^{2}}{20}=1$$

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(0,2)\)

Find parametric equations for the line with the given properties. Passing through \((12,7)\) and the origin

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$4 x^{2}-4 x-8 y+9=0$$

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$(7 x+24 y)^{2}=600 x-175 y+25$$

(a) Graph the conics $$r=\frac{e d}{(1+e \sin \theta)}$$ for \(e=1\) and various values of \(d .\) How does the value of \(d\) affect the shape of the conic? (b) Graph these conics for \(d=1\) and various values of \(e\) How does the value of \(e\) affect the shape of the conic?

Use a graphing device to graph the hyperbola. $$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$$

Use a graphing device to graph the ellipse. $$x^{2}+\frac{y^{2}}{12}=1$$

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F\left(0,-\frac{1}{2}\right)\)

Find parametric equations for the line with the given properties. Find parametric equations for the circle \(x^{2}+y^{2}=a^{2}\).

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$2 x^{2}-4 x y+2 y^{2}-5 x-5=0$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}+16=4\left(y^{2}+2 x\right)$$

Find an equation for the hyperbola that satisfies the given conditions. Foci \((\pm 5,0),\) vertices \((\pm 3,0)\)

Use a graphing device to graph the ellipse. $$6 x^{2}+y^{2}=36$$

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(-8,0)\)

Find parametric equations for the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$x^{2}-2 x y+3 y^{2}=8$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}-y^{2}=10(x-y)+1$$

Find an equation for the hyperbola that satisfies the given conditions. Foci \((0, \pm 10),\) vertices \((0, \pm 8)\)

Use a graphing device to graph the ellipse. $$x^{2}+2 y^{2}=8$$