The document defines an ellipse as the set of all points in a plane where the sum of the distances from two fixed points (the foci) is a constant. It provides the key elements and standard equations of an ellipse including the major and minor axes, eccentricity, foci, directrix, and equations when the major axis is parallel to the x-axis or y-axis. Examples are worked out sketching ellipses from their equations by completing the square to put them in standard form.

1. LECTURE UNIT 005
Ellipse
Set of all points in a plane such that the sum of the distances of each from two fixed point is a constant.
The fixed points are called the and the line through them is the
y
.
d d
Directrix
d4
Directrix
d3
a
Minor axis, 2b b
Latus rectum
c c
x
v2 F2 C (h, k) F1 v1
d2 b d1 Where:
F1, F2 = Focus
v1, v2 = Vertex
C = Center
a a
d1 + d2 = 2a
Major axis, 2a
Elements of Ellipse
a2 = b2 + c2
d3 c
Eccentricity, e = = a <1
d4
The eccentricity of a conic is a ratio of its distance from the focus
and from directrix.
a
d= e
2b2
Latus rectum, LR = a
Standard Equations
Where the major axis (M.A.) is parallel to the x-axis
(x - h)2 (y - k)2
+ =1
a2 b2
Where the major axis (M.A.) is parallel to the y-axis
(y - k)2 (x - h)2
+ =1
a2 b2
Note: a>b
Sketch the graph:
(x + 3)2 (y - 1)2
1. + =1
25 9
(y + 5)2 2
2. + (x - 1) = 1
25
(x + 2)2 y2
3. + =1
25 9
x2 y2
4. + =1
16 9
“True teaching is not an accumulation of knowledge; it is an
awakening of consciousness”

2. The equation Ax2 + By2+ Dx + Ey + F = 0 where A, B > 0 is the general equation of the ellipse. To sketch the
graph, reduce the equation to standard form.
(Ax2 + Dx) + (By2 + Ey) = - F
2 2 2 2
A x2 + D x + D 2
( ) + B (y 2
+ E y+ E2 D
) = -F + 4A + E
A 4A B 4B 4B
D2 2
M = -F + + E
4A 4B
2 2
( D
A x + 2A ) +B y+ E
( ) =M
2B
Hence;
2 2
D E
(x + 2A ) + (y + 2B )
=1
M M
A B
Note:
M < 0 no graph (imaginary ellipse)
M = 0 single point
M > 0 graph is an ellipse
D ,- E
c= -( 2A 2B )
5. 4x2 + 9y2 - 16x - 18y - 11 = 0
6. 4x2 + 8y2 - 16x - 16y + 24 = 0
7. 3x2 + 5y2 - 7x - y + 52 = 0
8. 25x2 + 9y2 + 100x + 18y + 16 = 0
9. 4x2 + 3y2 - 12 = 0
Find the equation of the parabola with given conditions.
10. With center at (3, 4), focus at (6, 4) and vertex at (8, 4).
11. With vertices at (-3, 7) and (-3, 3) and a focus at (-3, 6).
12. With center at the origin and passing through (-1, 3) and (2, 1).
“Seek peacefully, you will find”

3. ELLIPSE
Example 1:
(x + 3)2 (y - 1)2
+ =1
25 9
Solution: F F
C(-3, 1)
2 2
a = 25 b =9
a=5 b=3
Major Axis (M.A.) is parallel to the x-axis
C (-3, 1)
Example 8:
25x2 + 9y2 + 100x + 18y + 16 = 0
Solution:
(25x2 + 100x) + (9y2 + 18y) = -16
25(x2 + 4x) + 9(y2 + 2y) = -16
Completing squares;
25(x2 + 4x + 4) + 9(y2 + 2y + 1) = -16 + 100 + 9
25(x + 2)2 + 9(y + 1)2 = 93
(x + 2)2 (y + 1)2 C(-2, -1)
+ =1
3.8 10.3
a2 = 10.3 b2 = 3.8
a = 3.2 b = 1.9
Major Axis (M.A.) is parallel to the y-axis
C (-2, -1)
Example 10:
With center at (3, 4), focus at (6, 4) and vertex at (8, 4).
Solution:
Major Axis (M.A.) is parallel to the x-axis
(x + h)2 (y + k)2
+ =1
a2 b2
|CV1|: 8 - 3 = 5
|CF1|: 6 - 3 = 3
C(3, 4) F1(6, 4) V1(8, 4)
b = a2 - c 2
b = 52 - 32
b=4
Hence;
(x - 3)2 (y - 4)2
+ =1
25 16
“Never believe a word without putting its truth to the test”