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.
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