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

• 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.

More References and Links to Hyperbolas