Example 1: Find the points of intersection of a hyperbola and a line given by their equations as follows:
x^{2} / 9  y^{2} = 1
x + 5y = 3Solution 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  y^{2} = 1

Expand and group like terms and rewrite the equation as
16y^{2} 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.
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