Ellipses

1. Conic Sections

2. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E,
where A, B, C, D, and E are numbers (A or B ≠ 0).

3. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E,
where A, B, C, D, and E are numbers (A or B ≠ 0).
Their graphs are the conic sections as shown below.

4. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E,
where A, B, C, D, and E are numbers (A or B ≠ 0).
Their graphs are the conic sections as shown below.
Circles and ellipses
are enclosed.

5. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E,
where A, B, C, D, and E are numbers (A or B ≠ 0).
Their graphs are the conic sections as shown below.
Circles and ellipses
are enclosed.

6. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E,
where A, B, C, D, and E are numbers (A or B ≠ 0).
Their graphs are the conic sections as shown below.
If the equation Ax2 + By2 + Cx + Dy = E has A = B
so it’s of the form Ax2 + Ay2 + Cx + Dy = E,
Circles and ellipses
are enclosed.

7. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E,
where A, B, C, D, and E are numbers (A or B ≠ 0).
Their graphs are the conic sections as shown below.
If the equation Ax2 + By2 + Cx + Dy = E has A = B
so it’s of the form Ax2 + Ay2 + Cx + Dy = E,
dividing by A, we obtain 1x2 + 1y2 + #x + #y = #,
Circles and ellipses
are enclosed.

8. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E,
where A, B, C, D, and E are numbers (A or B ≠ 0).
Their graphs are the conic sections as shown below.
If the equation Ax2 + By2 + Cx + Dy = E has A = B
so it’s of the form Ax2 + Ay2 + Cx + Dy = E,
dividing by A, we obtain 1x2 + 1y2 + #x + #y = #,
and its graph is a circle.
Circles and ellipses
are enclosed.
Circles: 1x2 + 1y2 + #x + #y = #

9. Conic Sections
We continue with the graphs of Ax2 + By2 + Cx + Dy = E, where
A, B, C, D, and E are numbers (A or B ≠ 0).
Their graphs are the conic sections as shown below.
The graphs of Ax2 + Ay2 + Cx + Dy = E are circles.
Circles and ellipses
are enclosed.

10. Conic Sections
If an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,
but A and B having the same sign,
Circles and ellipses
are enclosed.
Circles: 1x2 + 1y2 + #x + #y = #

11. Conic Sections
If an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,
but A and B having the same sign, after dividing by A,
we obtain 1x2 + ry2 + #x + #y = #, with r > 0.
Circles and ellipses
are enclosed.
Circles: 1x2 + 1y2 + #x + #y = #

12. Conic Sections
If an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,
but A and B having the same sign, after dividing by A,
we obtain 1x2 + ry2 + #x + #y = #, with r > 0.
This is an ellipse.
Circles and ellipses
are enclosed.
Circles: 1x2 + 1y2 + #x + #y = #
Ellipses: 1x2 + ry2 + #x + #y = #
(r > 0)

13. Conic Sections
If an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,
but A and B having the same sign, after dividing by A,
we obtain 1x2 + ry2 + #x + #y = #, with r > 0.
This is an ellipse.
Geometrically, ellipses are “squashed/stretched” circles
along the horizontal/vertical directions of the circles.
Following is the distant–definition for ellipses.
Circles and ellipses
are enclosed.
Circles: 1x2 + 1y2 + #x + #y = #
Ellipses: 1x2 + ry2 + #x + #y = #
Ellipses also are
stretched or
compressed circles.
(r > 0)

14. Ellipses

15. Ellipses
Given two fixed points (called foci),
F2
F1

16. Ellipses
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
F2
F1

17. F2
F1
Ellipses
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.

18. F2
F1
P Q
R
( If P, Q, and R are any
points on an ellipse,
Ellipses
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.

19. F2
F1
P Q
R
p1
p2
( If P, Q, and R are any
points on an ellipse, then
p1 + p2
Ellipses
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.

20. F2
F1
P Q
R
p1
p2
( If P, Q, and R are any
points on an ellipse, then
p1 + p2
= q1 + q2
q1
q2
Ellipses
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.

21. F2
F1
P Q
R
p1
p2
( If P, Q, and R are any
points on an ellipse, then
p1 + p2
= q1 + q2
= r1 + r2
q1
q2
r2
r1
Ellipses
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.

22. F2
F1
P Q
R
p1
p2
( If P, Q, and R are any
points on an ellipse, then
p1 + p2
= q1 + q2
= r1 + r2
= a constant )
q1
q2
r2
r1
Ellipses
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.

23. F2
F1
P Q
R
p1
p2
( If P, Q, and R are any
points on an ellipse, then
p1 + p2
= q1 + q2
= r1 + r2
= a constant )
q1
q2
r2
r1
Ellipses
An ellipse also has a center (h, k );
(h, k) (h, k)
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.

24. F2
F1
P Q
R
p1
p2
( If P, Q, and R are any
points on an ellipse, then
p1 + p2
= q1 + q2
= r1 + r2
= a constant )
q1
q2
r2
r1
Ellipses
An ellipse also has a center (h, k ); it has two axes,
the semi-major (long)
(h, k)
Semi Major axis
(h, k)
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Semi Major axis

25. F2
F1
P Q
R
p1
p2
( If P, Q, and R are any
points on an ellipse, then
p1 + p2
= q1 + q2
= r1 + r2
= a constant )
q1
q2
r2
r1
Ellipses
An ellipse also has a center (h, k ); it has two axes,
the semi-major (long) and the semi-minor (short) axes.
(h, k)
Semi Major axis
(h, k)
Semi Minor axis
Given two fixed points (called foci), an ellipse is the set of
points whose sum of the distances to the foci is a constant.
Semi Major axis
Semi Minor axis

26. These semi-axes correspond to the important radii of the
ellipse.
Ellipses

27. These semi-axes correspond to the important radii of the
ellipse. From the center, the horizontal length is called the x-
radius
Ellipses
x-radius
x-radius

28. y-radius
These semi-axes correspond to the important radii of the
ellipse. From the center, the horizontal length is called the x-
radius and the vertical length the y-radius.
Ellipses
x-radius
x-radius
y-radius

29. These semi-axes correspond to the important radii of the
ellipse. From the center, the horizontal length is called the x-
radius and the vertical length the y-radius.
Ellipses
x-radius
The general equation for ellipses is
Ax2 + By2 + Cx + Dy = E
where A and B are the same sign but different numbers.
x-radius
y-radius
y-radius

30. These semi-axes correspond to the important radii of the
ellipse. From the center, the horizontal length is called the x-
radius and the vertical length the y-radius.
Ellipses
x-radius
The general equation for ellipses is
Ax2 + By2 + Cx + Dy = E
where A and B are the same sign but different numbers.
Using completing the square, such equations may be
transformed into the standard form of ellipses below.
x-radius
y-radius
y-radius

31. (x – h)2 (y – k)2
a2 b2
Ellipses
+ = 1
The Standard Form
(of Ellipses)

32. (x – h)2 (y – k)2
a2 b2
Ellipses
+ = 1 This has to be 1.
The Standard Form
(of Ellipses)

33. (x – h)2 (y – k)2
a2 b2
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
The Standard Form
(of Ellipses)

34. (x – h)2 (y – k)2
a2 b2
x-radius = a
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
The Standard Form
(of Ellipses)

35. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
The Standard Form
(of Ellipses)

36. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The Standard Form
(of Ellipses)

37. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The Standard Form
(of Ellipses)

38. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The x-radius is 4.
The Standard Form
(of Ellipses)

39. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2.
The Standard Form
(of Ellipses)

40. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
(3, -1)
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2.
The Standard Form
(of Ellipses)

41. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
(3, -1) (7, -1)
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2.
So the right point is (7, –1),
The Standard Form
(of Ellipses)

42. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
(3, -1) (7, -1)
(3, 1)
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2.
So the right point is (7, –1), the top
point is (3, 1),
The Standard Form
(of Ellipses)

43. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
(3, -1) (7, -1)
(-1, -1)
(3, -3)
(3, 1)
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2.
So the right point is (7, –1), the top
point is (3, 1), the left and bottom
points are (–1, –1) and (3, –3).
The Standard Form
(of Ellipses)

44. (x – h)2 (y – k)2
a2 b2
x-radius = a y-radius = b
(h, k) is the center.
Ellipses
+ = 1 This has to be 1.
(3, -1) (7, -1)
(-1, -1)
(3, -3)
(3, 1)
Example A. Find the center, major and minor axes.
Draw and label the top, bottom, right and left most points.
(x – 3)2 (y + 1)2
42 22
+ = 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2.
So the right point is (7, –1), the top
point is (3, 1), the left and bottom
points are (–1, –1) and (3, –3).
The Standard Form
(of Ellipses)

45. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Ellipses

46. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
Ellipses

47. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11
Ellipses

48. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11
Ellipses

49. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
Ellipses

50. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11
Ellipses

51. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11
+9
Ellipses

52. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11
+9 +16
Ellipses

53. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16
+9 +16
Ellipses

54. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16
+9 +16
Ellipses
9(x – 1)2 + 4(y – 2)2 = 36

55. 9(x – 1)2
4(y – 2)2
36 36
Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16
+9 +16
+ = 1
Ellipses
9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1

56. 9(x – 1)2
4(y – 2)2
36 4 36 9
Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16
+9 +16
+ = 1
Ellipses
9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1

57. 9(x – 1)2
4(y – 2)2
36 4 36 9
Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16
+9 +16
+ = 1
(x – 1)2 (y – 2)2
22 32
+ = 1
Ellipses
9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1

58. 9(x – 1)2
4(y – 2)2
36 4 36 9
Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16
+9 +16
+ = 1
(x – 1)2 (y – 2)2
22 32
+ = 1
Ellipses
9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
Hence, Center: (1, 2),
x-radius is 2,
y-radius is 3.

59. 9(x – 1)2
4(y – 2)2
36 4 36 9
Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the
standard form. Find the center and the x&y radii.
Draw and label the top, bottom, right, left most points.
Group the x’s and the y’s:
9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients
9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square
9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16
+9 +16
+ = 1
(x – 1)2 (y – 2)2
22 32
+ = 1
Ellipses
9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
Hence, Center: (1, 2),
x-radius is 2,
y-radius is 3.
(-1, 2) (3, 2)
(1, 5)
(1, -1)
(1, 2)

60. Conic Sections
Recall that after dividing the equations of ellipses
Ax2 + By2 + Cx + Dy = E by A,
we obtain 1x2 + ry2 + #x + #y = #, with r > 0.

61. Conic Sections
Recall that after dividing the equations of ellipses
Ax2 + By2 + Cx + Dy = E by A,
we obtain 1x2 + ry2 + #x + #y = #, with r > 0.
In the cases 1x2 + ry2 = 1where the ellipses centered at (0,0),
r controls the compression or extension factor
along the vertical or the y-direction of the circles
with the y-radius = 1/√r as shown.

62. Conic Sections
Recall that after dividing the equations of ellipses
Ax2 + By2 + Cx + Dy = E by A,
we obtain 1x2 + ry2 + #x + #y = #, with r > 0.
In the cases 1x2 + ry2 = 1where the ellipses centered at (0,0),
r controls the compression or extension factor
along the vertical or the y-direction of the circles
with the y-radius = 1/√r as shown.
r = 1
1x2 + 1y2 = 1
1x2 + y2 = 1
1
4
1x2 + y2 = 1
1
9
1x2 + 4y2 = 1
1x2 + 9y2 = 1
r = 4
r = 1/9
r = 1/4
r = 9
1
1 1 1
1
3
2
1/2
1
1/3

63. Ellipses

64. Ellipses
B. Complete the square of the following equations.
Find the center and the radii of the ellipses.
Draw and label the 4 cardinal points.
1. x2 + 4y2 = 1 2. 9x2 + 4y2 = 1
3. 4x2 + y2/9 = 1 4. x2/4 + y2/9 = 1
5. 0.04x2 + 0.09y2 = 1 6. 2.25x2 + 0.25y2 = 1
7. x2 + 4y2 = 100 8. x2 + 49y2 = 36
9. 4x2 + y2/9 = 9 10. x2/4 + 9y2 = 100
11. x2 + 4y2 + 8y = –3 12. y2 – 8x + 4x2 + 24y = 21
13. 4x2 – 8x + 25y2 + 16x = 71
14. 9y2 – 18y + 25x2 + 100x = 116

65. (Answers to odd problems) Exercise A.
1. + = 1
x2 y2
4 9
(2,0)
(0,3)
(0,-3)
(-2,0)
3. + = 1
(x + 1)2 (y + 3)2
4
16
(3,-3)
(-5,-3)
(-1,-5)
(-1,-1)
Ellipses
5. + = 1
(x + 4)2 (y + 2)2
16
1
(-3,-2)
(-5,-2)
(-4,-6)
(-4,2)
7. + = 1
(x + 1)2 (y – 2)2
3
2
(-1, 0.27)
(0.41, 2)
(-1,3.73)
(-2.47, 2)

66. 9. + = 1
(x – 3.1)2 (y + 2.3)2
0.09
1.44
Ellipses
(3.1, -2.6)
(3.1, -2)
(4.3, -2.3)
(1.9, -2.3)
Exercise B.
1. Center: (0,0)
x radius: 1
y radius: 0.5
3. Center: (0,0)
x radius: 0.5
y radius: 3
(0, -0.5)
(0, 0.5)
(-1, 0)
(0, -3)
(0, 3)
(0.5, 0)
(-0.5, 0)
(1, 0)

67. Ellipses
5. Center: (0,0)
x radius: 5
y radius: 10/3
(-5, 0)
(0, 3.33)
(0, -3.33)
(5, 0)
(0, 5)
(0, -5)
(-10, 0) (10, 0)
7. Center: (0,0)
x radius: 10
y radius: 5
(1.5, 0)
(-1.5, 0)
(0, 9)
(0, -9)
(1, -1)
(-1, -1)
(0, 0.5)
(0, -1.5)
9. Center: (0,0)
x radius: 1.5
y radius: 9
11. Center: (0,-1)
x radius: 1
y radius: 0.5
13. Center: (–1,0)
x radius: 18.75
y radius: 3
(–1,0)
(–1,–3)
(–1,3)
(–1+ 18.75,0)
(–1–18.75,0)