# 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. More links and references related to the above topics. Equation of Hyperbola- Graphing Problems.