first time posting and using the site. I have a quick problem that I need some help with. I need to find the equation of a hyperbola given the foci and the length of the minor axis.

The foci coordinates are as follows:

F(-5, 4) and F'(3, -2)

The length of the minor axis is 2√11.

Any help is very much appreciated.


I would recommend first applying a linear transformation that rotates the axis to become horizontal. If you plot the foci, you'll find that we need to rotate the axis counterclockwise an angle $\theta$, where $\cos\theta=4/5$. Here is the corresponding rotation matrix:


Under this rotation, the foci become $(-32/5,1/5)$ and $(18/5,1/5)$. Then, using the standard form of a hyperbola, we have the following equation:


Finally, we rotate our hyperbola back to its original orientation. The final result is:


or $2x^2+28x-24xy-5y^2-14y=133$.


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