Chapter 9 Problems

 (Chapter 10 starting with 8th edition)

Section 9.2         Section 9.3         Section 9.4

 (10.2 - 10.4 starting with 8th edition)

Section 9.2  (10.2 starting with 8th edition)

In problems 1-8 the graph of parabola is given. Match each graph to its equation.

6.  Vertex (0,0), opens in positive x direction, contains point (1,2) (graph is identical in 5th and 6th editions).

Find the equation of the parabola described. Find the two points that define the latus rectum.

10.Focus at (0,2); vertex at (0,0).

12.Focus at (-4,0); vertex at (0,0).

18.Vertex at (4, -2); focus at (6, -2).

20.Vertex at (0,0); axis of symmetry the x-axis; passing through the point (2,3)

24.focus at (-4,4); directrix the line y=-2

Find the vertex, focus, and directrix of each parabola. Graph the equation.

30.(x+4)^2=16(y+2)

36.x^2+6x-4y+1=0

40.x^2-4x=2y

42.y^2+12y= -x+1

Write an equation for parabola.  The figures are the same in both 5th and 6th editions.

48.  Vertex (1,-1), y-intercept (0,1), opens upward.

50.  Vertex (1,0), y-intercept (0,1), opens to left.

54.Constructing a headlight: A sealed-beam headlight is in the shape of the a paraboloid of revolution. The bulb, which is placed at the focus is 1 inch from the vertex. If the depth is to be 2 inches, what is the diameter of the headlight at its opening?

60.Reflecting telescopes: A reflecting telescope contains a mirror shaped like a paraboloid of revolution. If the mirror is 4 inches across at its opening and is 3 inches deep, where will the light collected be concentrated?

66.Show that the graph of an equation of the form

Cy^2+Dx+Ey+F=0, with C not 0

(a) Is a parabola if D is not 0

(b) Is a horizontal line if D=0 and E^2-4CF=0

(c) Is two horizontal lines if D=0 and E^2-4CF>0

(d) Contains no points if D=0 and E^2-4CF<0

Section 9.3  (10.3 starting with 8th edition)

Find the vertices and foci of each ellipse. Graph the equation.

6.x²/9 + y²/4 = 0

10.x²+9y²= 18

12.4y²+9x²=36

Find the equation for each ellipse. Graph the equation.

18.Center at (0,0); focus at (0,1); vertex at (0,-2)

20.Focus at (0, -4); vertices at (0,6 8)

24.Vertices at (6 5,0); c=2

28.  Find the equation of the ellipse pictured for problem 28 (figures in 5th and 6th editions are identical).

Find the center, foci, and vertices of each ellipse. Graph the equation.

30. (x-4)^2/9+(y+2)^2/4=1

36. 4x^2+3y^2+8x-6y=5

40. 9x^2+y^2-18x=0

Find the equation for each ellipse. Graph the equation.

42. Center at (-3,1); vertex at (-3,3); focus at(-3,0)

48. Center at (1,2); focus at (1,4); passing through the point (2,2)

50. Center at (1,2);vertex at (1,4); passing through the point (2,2)

Graph the function

54. f(x)= -`sqrt( (4-4x²))

60. Semielliptical arch bridge a bridge is built in the shape of semielliptical arch and is to have a span of 100 feet. The height of the arch at a distance of 40 feet from the center to be 10 feet. Find the height of the arch at its center.

64.  The mean distance of Mars from the sun is one 42 million miles and its perihelion is 28.5 million miles. What is the aphelian?  Note that full problem statement is the same in both the 5th and 6th editions.

66. Pluto The aphelion of Pluto is 4551million miles and the distance of the sun from the center of its illiptical orbit is 897.5 million miles. Find the aphelion of Pluto. What is the mean distance of Pluto from the sun? Write an equation for the orbit of Pluto about the sun.

70. Show that the graph of an equation of the form

Ax^2+Cy^2+Dx+Ey+F=0 A not 0, F not 0

Where A and C are of the same sign:

  1. Is an ellipse if (D^2/4A)+(E^2/4C)-F is the same as sign A
  2. Is a point if (D^2/4A)+(E^2/4C)-F=0.
  3. Contains no points if (D^2/4A)+(E^2/4C)-F is of opposite sign to A.

Section 9.4  (10.4 starting with 8th edition)

Find the equation for the hyperbola described. Graph the equation.

6. Center at (0,0); focus at (0,5); vertex at (0,3)

10. Focus at (0,6); vertices (0, -2) and (0,2)

12. Vertices at (0, -2) and (4,0); asymptote the line y=-x

Find the center, transverse axis, vertices, foci, and asymptotes. Graph the equation.

18. 4y²-x²=16

20. x²-y²=4

Write an equation for each hyperbola.

24.  See the figure for problem 26 in your text. 

Find an equation for the hyperbola described. Graph the equation.

30. Center at (1,4); focus at (-2,4); vertex at (0,4)

Find the center, transverse axis, vertices, foci, and asymptotes. Graph the equation.

36. (y+3)^2/4-(x-2)^2/9=1

40. (y-3)^2 - (x+2)^2=4

42. y^2-x^2-4y+4x-1=0

48. x^2-3y^2+8x-6y+4=0

Notice that each function is half a hyperbola.

50. F(x)= -`sqrt( (9+9x^2))

60. LORAN: two LORAN stations are positioned 100 miles apart along a straight shore.

  1. A ship records a time difference of 0.00032 seconds between the LORAN signals. Set up an appropriate rectangular coordinate system to determine where the ship would reach shore if it were to follow the hyperbola corresponding to this time difference.
  2. If the ship wants to enter a harbor located between the two stations 10 miles from the master station, what time difference should it be looking for?
  3. If the ship is 20 miles off shore when the desired time difference obtained, what is the approximate location of the ship?

[Note: the speed of each radio signal is 186,000 miles per hour.]

60. Prove that the hyperbola

y²/a²-x²/b²=1 has the two unique asymptotes

y= (a/b)x and y= (-a/b)x