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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