CONIC SECTIONS - ELLIPSES
Example 3:
Find the standard form of the equation of the ellipse having foci at (0, 1) and (4, 1) and a major axis
of length 6.
Graphing these two points in a coordinate system allows us to recognize that this must be an ellipse
with a horizontal major axis since the foci always lie on the major axis.
,
where
NOTE: Graphing is a MUST because it allows you to know which set of formulas to use!
Since the center of the ellipse lies exactly in between the two foci, we find it to be at (2, 1). Just look
at the graph!
Therefore, h = 2 and k = 1 and we can write
.
Next, we need to find the values of a and b.
We are told that the major axis is 6 units in length, that is, 2a = 6, and a = 3.
Lastly, we need to find b. Maybe the equation
can help us?
We know that the foci each lie c units from the center. Looking at the graph, we find c = 2.
The given information will now allow us to calculate the value for b.
Using the Square Root Property we find
.
Since we are talking about distances, b is strictly positive, therefore,
Therefore, the standard equation is
.
.