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 Two Hyperbolas
Find the points of intersection of the two hyperbolas given by their equations as follows:
\[
\frac{x^2}{4} - \frac{y^2}{16} = 1
\]
\[
\frac{(x - 1)^2}{2} - \frac{y^2}{4} = 1
\]
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:
\[
4x^2 - y^2 = 16
\]
\[
-2(x - 1)^2 + y^2 = -4
\]
- We now add the same sides of the two equations to obtain a quadratic equation
\[
4x^2 - 2(x - 1)^2 = 12
\]
- Expand and group like terms and rewrite the equation as
\[
2x^2 + 4x - 14 = 0
\]
- Solve the quadratic equation for x to obtain two solutions
\[
x \approx 1.83 \quad \text{and} \quad x \approx -3.83
\]
- We now substitute the values of x already obtained into the equation \(\frac{x^2}{4} - \frac{y^2}{16} = 1\) and solve it for y to obtain the y values
\[
\text{For } x \approx 1.83 \text{; there are real solutions for the equation } \frac{x^2}{4} - \frac{y^2}{16} = 1
\]
\[
\text{For } x \approx -3.83 \text{; } y \approx 6.53 \quad \text{and} \quad y \approx -6.53
\]
- The 2 points of intersection of the two hyperbolas are
\[
(-3.83, 6.53) \quad ; \quad (-3.83, -6.53)
\]
Shown below is the graph of two hyperbolas and their points of intersection.
More References and Links to Hyperbolas