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:
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![]() 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. |
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![]() 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 $x$-axis![]() Equation 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 $y$-axis![]() Equation 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. |
PROBLEM TYPES:
IN PROGRESS |