1. THE ELLIPSE
Week 17

2. THE DEFINITION
The locus of a point that moves in such a way that the ratio of its distances from a fixed point S (the focus) and from a fixed line ZL (the directrix) is constant (e) and less than 1

3. BASIC FEATURES The fixed line ZL is perpendicular to the x-axis
Let A divide SZ internally in the ratio e:1 (e<1)
Let A’ divide SZ externally in the ratio e:1 (e<1)

4. BASIC FEATURES
Thus, A and A’ are members of the loci of the ellipse (similar to the point P)
O is the origin and the mid-point of AA’

5. THE FOCUS Let AA’ = 2a SA = e. AZ SA’ = e. A’Z SA’ - SA = e (A’Z – AZ)
(OS+a) – (a-OS) = e(2a)
2.OS = 2.ae
OS = ae
The focus, S is the point with coordinates (-ae, 0)

6. EQUATION OF THE DIRECTRIX
SA = e. AZ
SA’ = e. A’Z
SA’ + SA = e (A’Z + AZ)
(2a) = e[(a+OZ) + (OZ-a)]
2a = 2e. OZ
OZ = a/e

7. THE EQUATION OF AN ELLIPSE

8. PROPERTIES OF AN ELLIPSE
Symmetrical about both axes
The foci are S (-ae, 0) and S’ (ae, 0)
The lines x = -a/e and x = a/e are the directrices

9. PROPERTIES OF AN ELLIPSE
AA’ = 2a (major axis)
BB’= 2b (minor axis)
O is the center
e is the eccentricity

10. ELLIPSE: ALTERNATIVE DEFINITION
An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points is constant.
The two fixed points are called the foci

11. EQUATION OF THE ELLIPSE
The equation of an ellipse can also be found by using the distance formula and the definition
An ellipse has foci at (5, 0) and (-5, 0). The distances from either of the x-intercepts to the foci are 2 units and 12 units. Find the equation of the ellipse.

12. EQUATION OF THE ELLIPSE
Consider the x-intercept, A.
AS+AS’=14 units
The sum of the distances from any point P(x, y) to the two foci must also be 14 units.
The distance formula can be used to find the equation of the ellipse
-5,0
5,0

13. EQUATION OF THE ELLIPSE

14. Properties of Ellipses
An ellipse is the set of all points in a plane such that the sum of the distances from the foci is constant
An ellipse has two axes of symmetry
The axis of the longer side of the ellipse is called the major axis and the axis of the shorter side is the minor axis
The focus points always lie on the major axis
The intersection of the two axes is the center of the ellipse
Major Axis
Focus
Center
Minor Axis
Focus

15. PROPERTIES OF THE ELLIPSE
The sum of the distances from any point to the foci = 2a (the length of the major axis).
The distance from the centre to either foci = c units

16. Important Information - Ellipses
Equation of the Ellipse
Foci Points
Is the major axis horizontal or vertical?
Center of the Ellipse
(x – h)2 + (y – k)2
a b2
( h + c, k) and (h – c, k)
Horizontal
(h, k)
b a2
(h, k + c) and (h, k – c)
Vertical
= 1
= 1
Important Notes:
In the above chart,
a2 > b2 always so a2 is always the larger number
If the a2 is under the x term, the ellipse is horizontal, if the a2 is under the y term the ellipse is vertical
You can tell that you are looking at an ellipse because: x2 is added to y2 and the x2 and y2 are divided by different numbers (if numbers were the same, it’s a circle)

17. Worked Example
Given an equation of an ellipse 16y2 + 9x2 – 96y – 90x = -225 find the coordinates of the center and foci as well as the lengths of the major and minor axis. Then draw the graph.
16 (y2 – 6y + o) + 9 (x2 – 10x + o) = (o) + 9(o)
16 (y2 – 6y + 9) + 9 (x2 – 10x + 25) = (9) + 9(25)
16 (y – 3)2 + 9 (x – 5)2 = 144
(y – 3)2 + (x – 5)2
= 1
Center: (5, 3)
16 > 9 so the foci are on the horizontal axis
c = – 9
c = 7
Foci: ( , 3) and (5 – 7, 3)
Major Axis Length = 4 (2) = 8
Minor Axis Length =

18. Sample Problems
For 49x2 + 16y2 = 784 find the center, the foci, and the lengths of the major and minor axes. Then draw the graph.
Write an equation for an ellipse with foci (4, 0) and (-4, 0). The endpoints of the minor axis are (0, 2) and (0, -2).
X2 + y2
Foci: (0, - 33) (0, 33) Center: (0, 0) Length of major= 14 Length of minor= 8
= 1 #1