The Ellipse
An ellipse is the set of points P in the plane such that the sum of the
distances from P to two fixed points F_{1} and F_{ 2} is a
constant.
F_{1}P + F_{ 2}P = k the points F_{1} & F_{ 2}
the foci.
Using the idea of the two stacked cones, the ellipse is made when an angled
vertical cut is made
The important characteristics of the ellipse:
Example 1
Given 4x^{ 2} + y^{ 2} = 36 , find the equation of the
ellipse and give the coordinates of the foci.
Solution:
The foci
Given the equation, we see the major axis is along the y axis, and is centred
at the origin.
This means the vertices are at (0, 6) & (0,6), with co vertices at (3,0) &
(3,0).
a^{ 2 } 
= b^{ 2} + c^{ 2} 
36  9 
= c^{ 2} 
27 
= c^{ 2} 

= c 
Hence the foci are at (0,
) and (0, )
Ellipse are often not centered at the origin: Hence the possible ellipses are
Sketch
, and label the
foci
We can from the equation that the major axis is parallel to the x axis.
The length of the major axis is 2a, which is 10. the minor axis is 2b, which is
8.
The vertices are (h  a , k) & (h + a, k), which are going to give ( 6, 3) &
(4, 3).
The covertices are (h, k  b) & ( h, k + b), which are going to give (1, 1) &
(1, 7).
The foci are going to be (h  c, k) & ( h + c, k)
Where c is equal to
a^{ 2 } 
= b^{ 2} + c^{ 2} 
c^{ 2} 
= 25  16 
c 
= 3 
Hence the foci are ( 4, 3 ) & ( 2, 3 )
Write the equation of an ellipse in standard form given:
Â· The centre is at (2,1).
Â· The major axis is 16 units and is parallel to the x axis.
Â· The minor axis is 4 units.
Solution:
The centre is at (2,1), with h = 2 & k = 1.
If the length of the major axis is 16, and the major axis is 2a, then
16 = 2a
a = 8
The length of the minor axis is 2.
Since we know h, k, a ,b we can solve the equation.
