Hyperbolas were introduced in two prior lessons:
In this current section, a hyperbola is positioned with its center at the origin and foci on the $x$axis or $y$axis.
Consider a hyperbola with foci $\,F_1\,$ and $\,F_2\,$ and hyperbola constant $\,k\,.$ Recall from the previous section that:

The center of the hyperbola is the midpoint of the line segment connecting the foci; it is shown in red. The minor axis of the hyperbola is the line through the center that is perpendicular to the major axis; it is dashed purple. 

and center at the origin $c := \,$ distance from center to each focus $a := \,$ distance from center to each vertex Note that $\,0 < a < c\,.$ 
The derivation of the equation of a hyperbola with center at the origin and
foci on the $y$axis is nearly identical to the derivation above.
Foci on the $x$axisEquation of Hyperbola: $$\cssId{s172}{\frac{x^2}{a^2}  \frac{y^2}{b^2} = 1}$$ When the foci are on the $\color{red}{x}$axis, the $\,\color{red}{x^2}$term is positive. Coordinates of foci: $\,(c,0)\,$ and $\,(c,0)\,$ 
Foci on the $y$axisEquation of Hyperbola: $$\cssId{s179}{\frac{y^2}{a^2}  \frac{x^2}{b^2} = 1}$$ When the foci are on the $\color{red}{y}$axis, the $\,\color{red}{y^2}$term is positive. Coordinates of foci: $\,(0,c)\,$ and $\,(0,c)\,$ 
Find the equation of the following hyperbola:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
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