Review
A hyperbola with center at the origin (0,0), is the graph
of
The graph of the equation on the left has the
following properties: x intercepts at ± a , no y intercepts, foci at (c , 0)
and (c , 0), asymptotes with equations y = ± x (b/a). The graph of
the equation on the right has the following properties: y intercepts at ± a , no x intercepts, foci at (0 , c)
and (0 , c), asymptotes with equations y = ± x (a/b) .
a, b and c are related by c^{2} = a^{2}
+ b^{2}.
The length of the transverse axis is 2a, and the length of
the conjugate axis is 2b.
Applying the symmetry tests for graphs of equations in two
variables, the hyperbola is symmetric with respect to x axis, y axis and the
origin.
A tutorial on the definiton and properties of hyperbolas can be found in this site.
Problem 1: Given the following equation
9x^{2}  16y^{2} = 144
a) Find the x and y intercepts, if possible, of the graph of the equation.
b) Find the coordinates of the foci.
c) Sketch the graph of the equation.
Solution to Problem1
a) We first write the given equation in standard form
by dividing both sides of the equation by 144
9x^{2} / 144  16y^{2} / 144 = 1
x^{2} / 16  y^{2} / 9 = 1
x^{2} / 4^{2}  y^{2} / 3^{2} =
1
We now compare the equation obtained with the
standard equation (left) in the review above and we can say that the given equation is
that of an hyperbola with a = 4 and b = 3.
Set y = 0 in the equation obtained and find the x
intercepts.
x^{2} / 4^{2} = 1
Solve for x.
x^{2} = 4^{2}
x^{ } = ± 4
Set x = 0 in the equation obtained and find the y
intercepts.

y^{2} / 3^{2} = 1
NO y intercepts since the above equation
does not have real solutions.
b) We need to find c first.
c^{2} = a^{2} + b^{2}
a and b were found in part a).
c^{2} = 4^{2} + 3^{2}
c^{2} = 25
Solve for c.
c = ± 5
The foci are F_{1}
(5 , 0)
and F_{2} (5 , 0)
c)
1  Find the asymptotes y = 
(b/a) x and y = (b/a) x and
plot them.
y = (3/4) x and y = (3/4) x
2  plot
the x intercepts
3  Find extra points (if necessary)
set x = 6 and find
y
9(6)^{2}  16y^{2} = 144
 16y^{2} = 144  324
y^{2} = 45 / 4
Solve for
y
y = 3(5)^{1/2} / 2
and y =  3(5)^{1/2}
/ 2
so the points (6, 3(5)^{1/2} / 2)
and (6, 3(5)^{1/2} / 2) are on the graph of the
hyperbola.
Also because of the symmetry of the graph of the
hyperbola, the points (6, 3(5)^{1/2} / 2)
and
(6, 3(5)^{1/2} / 2) are also on the graph
of the hyperbola.
Matched Problem: Given the following equation
x^{2}  y^{2} = 9
a) Find the x and y intercepts, if possible, of the graph of the equation.
b) Find the coordinates of the foci.
c) Sketch the graph of the equation.
