Find the Points of Intersection of Two Hyperbolas






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:

x2 / 4 - y2 / 16 = 1
(x - 1)2 / 2 - y2 / 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 x2 - y2 = 16
    - 2 (x - 1)2 + y2 = - 4


  • We now add the same sides of the two equations to obtain a quadratic equation
    4 x2 - 2 (x - 1)2 = 12

  • Expand and group like terms and rewrite the equation as
    2 x2 + 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 x2 / 4 - y2 / 16 = 1 and solve it for y to obatain the y values

    for x ≈ 1.83 ; there are real solutions for the equation x2 / 4 - y2 / 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.

Points of intersection of two hyperbola



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