This is a tutorial with detailed solutions to problems related to the ellipse equation. For an interactive tutorial on the equation of the ellipse Go here.
Review
An ellipse with center at the origin (0,0), is the graph
of
with a > b > 0
The length of the major axis is 2a, and the length of the
minor axis is 2b. The two foci (foci is the plural of focus) are at (± c , 0)
or at (0 , ± c), where c^{2} = a^{2}  b^{2}.
Problem 1: Given the following equation
9x^{2} + 4y^{2} = 36
a) Find the x and y intercepts of the graph of the equation.
b) Find the coordinates of the foci.
c) Find the length of the major and minor axes.
d) Sketch the graph of the equation.
Solution to Example 1
a) We first write the given equation in standard form
by dividing both sides of the equation by 36
9x^{2} / 36 + 4y^{2} / 36 = 1
x^{2} / 4 + y^{2} / 9 = 1
x^{2} / 2^{2} + y^{2} / 3^{2} = 1
We now identify the equation obtained with one of the
standard equation in the review above and we can say that the given equation is
that of an ellipse with a = 3 and b = 2 (NOTE: a
>b) .
Set y = 0 in the equation obtained and find the x
intercepts.
x^{2} / 2^{2} = 1
Solve for x.
x^{2} = 2^{2}
x = ± 2
Set x = 0 in the equation obtained and find the y
intercepts.
y^{2} / 3^{2} = 1
Solve for y.
y^{2} = 3^{2}
y = ± 3
b) We need to find c first.
c^{2} = a^{2}  b^{2}
a and b were found in part a).
c^{2} = 3^{2}  2^{2}
c^{2} = 5
Solve for c.
c = ± (5)^{1/2}
The foci are F_{1} (0 , (5)^{1/2})
and F_{2} (0 , (5)^{1/2})
c) The major axis length is given by 2
a = 6.
The minor axis length is given by
2 b = 4.
d) Locate the x and y intercepts, find extra points if
needed and sketch.
Matched Problem: Given the following
equation
4x^{2} + 9y^{2} = 36
a) Find the x and y intercepts of the graph of the equation.
b) Find the coordinates of the foci.
c) Find the length of the major and minor axes.
d) Sketch the graph of the equation.
