Orienting the Hyperbola

Vertical Orientation

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A hyperbola is in the vertical position if the transverse axis is vertical. The standard form of a vertical hyperbola with center \((h, k)\) is:

\(\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1\)

Note that the term with the y-variable is positive while that with the x-variable is negative. Furthermore, \(a_2\) is the denominator of the positive term.

We can expand this standard form to obtain its general form:

\(A x^2+B y^2+C x+D y+E=0\)

Like the horizontal hyperbola, it is a polynomial of second degree. In this case, \(A>0\) and \(B<0\) for the vertical hyperbola.

Summary

A hyperbola can have three orientations: vertical, horizontal, or inclined. We base these orientations on the orientation of the hyperbola's transverse axis.

A hyperbola is said to be horizontal if its transverse axis is horizontal. If we have a horizontal hyperbola with center \((h, k)\), the standard equation is: \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\)

A hyperbola is in the vertical position if its transverse axis is vertical. The standard form of a vertical hyperbola with center \((h, k)\) is: \(\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1\)

The general form of both orientations is \(A x^2+B y^2+C x+D y+E=0\). It is a horizontal hyperbola if \(A<0\) and \(B>0\). If \(A>0\) and \(B<0\), then it is vertical.