
This is tutorial on finding the points of intersection of two hyperbolas given by their equations.
Example 1
Find the points of intersection of the two hyperbolas given by their equations as follows:
x^{2} / 4  y^{2} / 16 = 1
(x  1)^{2} / 2  y^{2} / 4 = 1
Solution to Example 1

We first multiply all terms of the first equation by 16 and all the terms of the second equation by  4 to obtain equivalent equations:
4 x^{2}  y^{2} = 16
 2 (x  1)^{2} + y^{2} =  4

We now add the same sides of the two equations to obtain a quadratic equation
4 x^{2}  2 (x  1)^{2} = 12

Expand and group like terms and rewrite the equation as
2 x^{2} + 4x  14 = 0

Solve the quadratic equation for x to obtain two solutions
x ≈ 1.83 and x ≈ 3.83

We now substitute the values of x already obtained into the equation x^{2} / 4  y^{2} / 16 = 1 and solve it for y to obatain the y values
for x ≈ 1.83 ; there are real solutions for the equation x^{2} / 4  y^{2} / 16 = 1
for x ≈ 3.83 ; y values are given by: y ≈ 6.53
and y ≈ 6.53

The 2 points of intersection of the two hyperbolas are
( 3.83 , 6.53) ; ( 3.83 , 6.53)
Shown below is the graph of two hyperbolas and their points of intersection.
More References and Links to HyperbolasEquation of Hyperbola Graphing Problems.hyperbola equation
Find the Points of Intersection of Two Hyperbolas
Points of Intersection of a Hyperbola and a Line
