Solution: 5560. Polar equations for conic sections Graph the following conic sections

Give the property that defines all parabolas.

Give the property that defines all ellipses.

Give the property that defines all hyperbolas.

Sketch the three basic conic sections in standard position with vertices and foci on the x-axis.

Sketch the three basic conic sections in standard position with vertices and foci on the y-axis.

What is the equation of the standard parabola with its vertex at the origin that opens downward?

What is the equation of the standard ellipse with vertices at 1{a, 02 and foci at 1{c, 02?

What is the equation of the standard hyperbola with vertices at 10, {a2 and foci at 10, {c2?

Given vertices 1{a, 02 and eccentricity e, what are the coordinates of the foci of an ellipse and a hyperbola? 1

Give the equation in polar coordinates of a conic section with a focus at the origin, eccentricity e, and a directrix x = d, where d 7 0. 1

What are the equations of the asymptotes of a standard hyperbola with vertices on the x-axis? 1

How does the eccentricity determine the type of conic section? 1

1318. Graphing parabolas Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. x2 = 12y 1

1318. Graphing parabolas Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. y2 = 20x 1

1318. Graphing parabolas Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. x = -y2>16 1

1318. Graphing parabolas Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. 4x = -y2 1

1318. Graphing parabolas Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. 8y = -3x2 1

1318. Graphing parabolas Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. 12x = 5y2 1

1924. Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola that opens to the right with directrix x = -4 2

1924. Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola that opens downward with directrix y = 6 2

1924. Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola with focus at 13, 02 2

1924. Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola with focus at 1-4, 02 2

1924. Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola symmetric about the y-axis that passes through the point 12, -62 2

1924. Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola symmetric about the x-axis that passes through the point 11, -42 2

2526. From graphs to equations Write an equation of the following parabolas.

2526. From graphs to equations Write an equation of the following parabolas.

2732. Graphing ellipses Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. x2 4 + y2 = 1 2

2732. Graphing ellipses Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. x2 9 + y2 4 = 1 2

2732. Graphing ellipses Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. x2 4 + y2 16 = 1 3

2732. Graphing ellipses Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. x2 + y2 9 = 1 3

2732. Graphing ellipses Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. x2 5 + y2 7 = 1 3

2732. Graphing ellipses Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. 12x2 + 5y2 = 60 3

3336. Equations of ellipses Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse whose major axis is on the x-axis with length 8 and whose minor axis has length 6 3

3336. Equations of ellipses Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse with vertices 1{6, 02 and foci 1{4, 02 3

3336. Equations of ellipses Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse with vertices 1{5, 02, passing through the point 14, 35 2 3

3336. Equations of ellipses Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse with vertices 10, {102, passing through the point 113>2, 52 3

3738. From graphs to equations Write an equation of the following ellipses.

3738. From graphs to equations Write an equation of the following ellipses.

3944. Graphing hyperbolas Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. x2 4 - y2 = 1 4

3944. Graphing hyperbolas Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. y2 16 - x2 9 = 1 4

3944. Graphing hyperbolas Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. 4x2 - y2 = 16 4

3944. Graphing hyperbolas Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. 25y2 - 4x2 = 100 4

3944. Graphing hyperbolas Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. x2 3 - y2 5 = 1 4

3944. Graphing hyperbolas Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. 10x2 - 7y2 = 140 4

4548. Equations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices 1{4, 02 and foci 1{6, 02 4

4548. Equations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices 1{1, 02 that passes through 153 , 82 4

4548. Equations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices 1{2, 02 and asymptotes y = {3x>2 4

4548. Equations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices 10, {42 and asymptotes y = {2x 4

4950. From graphs to equations Write an equation of the following hyperbolas.

4950. From graphs to equations Write an equation of the following hyperbolas.

5154. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. An ellipse with vertices 1{9, 02 and eccentricity 13 5

5154. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. An ellipse with vertices 10, {92 and eccentricity 14 5

5154. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. A hyperbola with vertices 1{1, 02 and eccentricity 3 5

5154. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. A hyperbola with vertices 10, {42 and eccentricity 2 5

5560. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. r = 4 1 + cos u 5

5560. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. r = 4 2 + cos u 5

5560. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. r = 1 2 - cos u 5

5560. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. r = 6 3 + 2 sin u 5

5560. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. r = 1 2 - 2 sin u 6

5560. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. r = 12 3 - cos u 6

6164. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2p. r = 1 1 + sin u 6

6164. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2p. r = 1 1 + 2 cos u 6

6164. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2p. r = 3 1 - cos u 6

6164. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2p. r = 1 1 - 2 cos u 6

Parabolas with a graphing utility Use a graphing utility to graph the parabolas y2 = 4px, for p = -5, -2, -1, 1, 2, and 5 on the same set of axes. Explain how the shapes of the curves vary as p changes. 6

Hyperbolas with a graphing utility Use a graphing utility to graph the hyperbolas r = e 1 + e cos u , for e = 1.1, 1.3, 1.5, 1.7, and 2 on the same set of axes. Explain how the shapes of the curves vary as e changes. 6

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The hyperbola x2>4 - y2>9 = 1 has no y-intercepts. b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number. c. Given the directrices and foci of a standard hyperbola, it is possible to find its vertices, eccentricity, and asymptotes. d. The point on a parabola closest to the focus is the vertex. 6

6871. Tangent lines Find an equation of the line tangent to the following curves at the given point. y2 = 8x; 18, -82 6

6871. Tangent lines Find an equation of the line tangent to the following curves at the given point. x2 = -6y; 1-6, -62 7

6871. Tangent lines Find an equation of the line tangent to the following curves at the given point. r = 1 1 + sin u ; a 2 3 , p 6 b 7

6871. Tangent lines Find an equation of the line tangent to the following curves at the given point. y2 - x2 64 = 1; a6, - 5 4 b 7

7273. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

7273. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

Deriving polar equations for conics Modify Figure 10.57 to derive the polar equation of a conic section with a focus at the origin in the following three cases. a. Vertical directrix at x = -d, where d 7 0 b. Horizontal directrix at y = d, where d 7 0 c. Horizontal directrix at y = -d, where d 7 0 7

Another construction for a hyperbola Suppose two circles, whose centers are at least 2a units apart (see figure), are centered at F1 and F2, respectively. The radius of one circle is 2a + r and the radius of the other circle is r, where r 0. Show that as r increases, the intersection point P of the two circles describes one branch of a hyperbola with foci at F1 and F2. Hyperbola 2a _ r r F1 F2 P1 P2 7

The ellipse and the parabola Let R be the region bounded by the upper half of the ellipse x2>2 + y2 = 1 and the parabola y = x2>12. a. Find the area of R. b. Which is greater, the volume of the solid generated when R is revolved about the x-axis or the volume of the solid generated when R is revolved about the y-axis? 7

Tangent lines for an ellipse Show that an equation of the line tangent to the ellipse x2>a2 + y2>b2 = 1 at the point 1x0, y02 is xx0 a2 + yy0 b2 = 1. 7

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x2>a2 - y2>b2 = 1 at the point 1x0, y02. 7

Volume of an ellipsoid Suppose that the ellipse x2>a2 + y2>b2 = 1 is revolved about the x-axis. What is the volume of the solid enclosed by the ellipsoid that is generated? Is the volume different if the same ellipse is revolved about the y-axis? 8

Area of a sector of a hyperbola Consider the region R bounded by the right branch of the hyperbola x2>a2 - y2>b2 = 1 and the vertical line through the right focus. a. What is the area of R? b. Sketch a graph that shows how the area of R varies with the eccentricity e, for e 7 1. 8

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x2>a2 - y2>b2 = 1 and the vertical line through the right focus. a. What is the volume of the solid that is generated when R is revolved about the x-axis? b. What is the volume of the solid that is generated when R is revolved about the y-axis? 8

Volume of a paraboloid (Archimedes) The region bounded by the parabola y = ax2 and the horizontal line y = h is revolved about the y-axis to generate a solid bounded by a surface called a paraboloid (where a 7 0 and h 7 0). Show that the volume of the solid is 32 the volume of the cone with the same base and vertex. (See the Guided Project Properties of Conic sections for additional applications of conic sections.) 8

Reflection property of parabolas Consider the parabola y = x2>4p with its focus at F 10, p2 (see figure). The goal is to show that the angle of incidence between the ray / and the tangent line L (a in the figure) equals the angle of reflection between the line PF and L (b in the figure). If these two angles are equal, then the reflection property is proved because / is reflected through F. a. Let P1x0, y02 be a point on the parabola. Show that the slope of the line tangent to the curve at P is tan u = x0>12p2. b. Show that tan w = 1p - y02>x0. c. Show that a = p>2 - u; therefore, tan a = cot u. d. Note that b = u + w. Use the tangent addition formula tan 1u + w2 = tan u + tan w 1 - tan u tan w to show that tan a = tan b = 2p>x0. e. Conclude that because a and b are acute angles, a = b. O P L x _ _ _ _ _ F(0, p) x2 4p y _ _ y y0 x0 8

Golden Gate Bridge Completed in 1937, San Franciscos Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 8

Equation of an ellipse Consider an ellipse to be the set of points in a plane whose distances from two fixed points have a constant sum 2a. Derive the equation of an ellipse. Assume the two fixed points are on the x-axis equidistant from the origin. 8

Equation of a hyperbola Consider a hyperbola to be the set of points in a plane whose distances from two fixed points have a constant difference of 2a or -2a. Derive the equation of a hyperbola. Assume the two fixed points are on the x-axis equidistant from the origin. 8

Equidistant set Show that the set of points equidistant from a circle and a line not passing through the circle is a parabola. Assume the circle, line, and parabola lie in the same plane. 8

Polar equation of a conic Show that the polar equation of an ellipse or a hyperbola with one focus at the origin, major axis of length 2a on the x-axis, and eccentricity e is r = a11 - e22 1 + e cos u . 8

Shared asymptotes Suppose that two hyperbolas with eccentricities e and E have perpendicular major axes and share a set of asymptotes. Show that e-2 + E-2 = 1. 9

The lines tangent to the endpoints of any focal chord of a parabola y2 = 4px intersect on the directrix and are perpendicular. 9

Let L be the latus rectum of the parabola y2 = 4px, for p 7 0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D + _FP_ is a constant. Find the constant. 9

The length of the latus rectum of the parabola y2 = 4px or x2 = 4py is 4_ p _. 9

The length of the latus rectum of an ellipse centered at the origin is 2b2>a = 2b21 - e2. 9

The length of the latus rectum of a hyperbola centered at the origin is 2b2>a = 2b2e2 - 1. 9

Confocal ellipse and hyperbola Show that an ellipse and a hyperbola that have the same two foci intersect at right angles. 9

Approach to asymptotes Show that the vertical distance between a hyperbola x2>a2 - y2>b2 = 1 and its asymptote y = bx>a approaches zero as xS _, where 0 6 b 6 a. 9

Sector of a hyperbola Let H be the right branch of the hyperbola x2 - y2 = 1 and let / be the line y = m1x - 22 that passes through the point 12, 02 with slope m, where - _ 6 m 6 _. Let R be the region in the first quadrant bounded by H and / (see figure). Let A1m2 be the area of R. Note that for some values of m, A1m2 is not defined. a. Find the x-coordinates of the intersection points between H and / as functions of m; call them u1m2 and v1m2, where v1m2 7 u1m2 7 1. For what values of m are there two intersection points? b. Evaluate lim mS1+ u1m2 and lim mS1+ v1m2. c. Evaluate lim mS_ u1m2 and lim mS_ v1m2. d. Evaluate and interpret lim mS_ A1m2. _: y _ m(x _ 2) H: x2 _ y2 _ 1 R y v(m) x u(m) Area _ A(m) 1 2 9

The anvil of a hyperbola Let H be the hyperbola x2 - y2 = 1 and let S be the 2-by-2 square bisected by the asymptotes of H. Let R be the anvil-shaped region bounded by the hyperbola and the horizontal lines y = {p (see figure). a. For what value of p is the area of R equal to the area of S? b. For what value of p is the area of R twice the area of S? 2 2 x H: x2 _ y2 _ 1 S y _ p y _ _p Anvil y 9

Parametric equations for an ellipse Consider the parametric equations x = a cos t + b sin t, y = c cos t + d sin t, where a, b, c, and d are real numbers. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form Ax2 + Bxy + Cy2 = K, where A, B, C, and K are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the x- and y-axes provided ab + cd = 0. c. Show that the equations describe a circle provided ab + cd = 0 and c2 + d2 = a2 + b2 _ 0.