Points of Intersection of a Hyperbola and a Line




This is tutorial on finding the points of intersection of a hyperbola and a line given by their equations.

Example 1: Find the points of intersection of a hyperbola and a line given by their equations as follows:

x2 / 9 - y2 = 1
x + 5y = 3

Solution to Example 1:

  • Solve the equation of the line for x to obtain:
    x = 3 - 5y

  • We now substitute x by 3 - 5y into the equation of the hyperbola to obtain
    (3 - 5y)2 / 9 - y2 = 1

  • Expand and group like terms and rewrite the equation as
    16y2 -30y = 0

  • Solve the quadratic equation for y to obtain two solutions

    y = 0 and y = 15/8

  • We now substitute the values of y already obtained into x = 3 - 5y to obtain

    for y = 0, x = 3 and for y = 15/8, x = -51/8

  • The 2 points of intersection of the the hyperbola and the line are

    ( 3 , 0) ; ( -51/8 , 15/8)

Shown below is the graph of a hyperbola, a line and their points of intersection.

Points of intersection of a hyperbola and a line



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