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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.

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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:

$$\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}=\begin{pmatrix}\frac{4}{5}&-\frac{3}{5}\\\frac{3}{5}&\frac{4}{5}\end{pmatrix}$$

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:

$$\frac{(x+\frac{7}{5})^2}{14}-\frac{(y-\frac{1}{5})^2}{11}=1$$

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

$$\frac{(\frac{4x}{5}-\frac{3y}{5}+\frac{7}{5})^2}{14}-\frac{(\frac{3x}{5}+\frac{4y}{5}-\frac{1}{5})^2}{11}=1$$

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

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