# Thread: How to Determine a Hyperbola From Points

1. ## How to Determine a Hyperbola From Points

Hi,
Is there a way to determine a hyperbola (equation) from a few points (three maybe)?

I thought that there is only one way, which is by graphing and finding the vertex, transverse axes, etc... but I hope there are other ways to determine and equation.

Thanks!

2. Originally Posted by masoug
Hi,
Is there a way to determine a hyperbola (equation) from a few points (three maybe)?

I thought that there is only one way, which is by graphing and finding the vertex, transverse axes, etc... but I hope there are other ways to determine and equation.

Thanks!
If you have the general equation of a hyperbola a set of points defines a set of simultaneous equations in the parameters which can then be solved.

CB

3. Originally Posted by CaptainBlack
If you have the general equation of a hyperbola a set of points defines a set of simultaneous equations in the parameters which can then be solved.

CB
Um, can you explain what you mean by that?

4. If you know that a hyperbolas axes are parallel to the x and y axes, then a hyperbola can be written as either
$\displaystyle \frac{(x- x_0)^2}{a^2}- \frac{y- y_0)^2}{b^2}= 1$
or
$\displaystyle \frac{(y- y_0)^2}{b^2}- \frac{(x- x_0)^2}{a^2}=$

That depends on 4 parameters, $\displaystyle x_0$, $\displaystyle y_0$, a, and b. If you know 4 points you can put the x,y values of those points into the equation, getting 4 equations to solve for the parameters.

If it is possible that the axes of the hyperbola are tilted with respect to the x and y axes the problem becomes much more complicated. However, any conic section (hyperbola, parabola, ellipse, circle and some special cases) can be written in the form $\displaystyle Ax^2+ Bxy+ Cy^2+ Dx+ Ey+ F= 0$. That has 6 parameters but we could always divide the entire equation by one of the so there are 5 independent parameters. That means that 5 points are sufficient to determine any conic section.

5. ## Aha!

Originally Posted by HallsofIvy
If it is possible that the axes of the hyperbola are tilted with respect to the x and y axes the problem becomes much more complicated. However, any conic section (hyperbola, parabola, ellipse, circle and some special cases) can be written in the form $\displaystyle Ax^2+ Bxy+ Cy^2+ Dx+ Ey+ F= 0$. That has 6 parameters but we could always divide the entire equation by one of the so there are 5 independent parameters. That means that 5 points are sufficient to determine any conic section.
I see...
So for most conic sections, just use the $\displaystyle Ax^2+ Bxy+ Cy^2+ Dx+ Ey+ F= 0$ relation and pick six points to satisfy each unknown.

Thanks!

6. For all conic sections and 5 points will be sufficient.