Hyperbolas
Hyperbolas
Horizontal transverse axis
\[\dfrac{(x-a)^2}{p^2}-\dfrac{(y-b)^2}{q^2}=1\]
Vertical transverse axis
\[\dfrac{(y-b)^2}{q^2}-\dfrac{(x-a)^2}{p^2}=1\]
Finding the oblique asymptotes
Set the right-hand side to zero: \(\dfrac{(x-a)^2}{p^2}-\dfrac{(y-b)^2}{q^2}=0\), or \(\dfrac{(y-b)^2}{q^2}-\dfrac{(x-a)^2}{p^2}=0\), and make \(y\) the subject.
As \(x\to\infty\), \(\dfrac{(x-a)^2}{p^2}\approx\dfrac{(y-b)^2}{q^2}\). Hence the asymptotes are \(y=b\pm\dfrac{q}{p}(x-a)\).
Gradients of the oblique asymptotes: \(m=\pm\dfrac{q}{p}\).
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