## Equation of Hyperbola From Asymptotes and Point on Curve

A hyperbola has asymptotes\[x=\frac{3}{2}, \: y=4\]

and passes through the point \[(2,1)\]

.How can we find the equation of the asymptote?

The equation of a hyperbola can be written in the form

\[(x-x_0)(y-y_0)=k\]

where \[x=x_0, \: y=y_0\]

are the equations of the hyperbolae and \[k\]

is a constant.Hence we can write

\[(x- \frac{3}{2})(y-4)=k\]

.To find the value of

\[k\]

substitute the equation of a point on the curve.\[((2- \frac{3}{2})(1-4)=k \rightarrow k=- \frac{3}{2}\]

.The equation of the hyperbola is

\[(x- \frac{3}{2})(y-4)=- \frac{3}{2}\]

.We can write this as

\[y-4=\frac{-3/2}{x-3/2} \rightarrow y= \frac{-3/2}{x-3/2} +4=\frac{-3/2+4(x-3/2)}{x-3/2}=\frac{4x-15/2}{x-3/2}=\frac{8x-15}{2x-3}\]

We can write this as

\[y-4=\frac{-3/2}{x-3/2} \rightarrow y= \frac{-3/2}{x-3/2} +4=\frac{-3/2+4(x-3/2)}{x-3/2}=\frac{4x-15/2}{x-3/2}=\frac{8x-15}{2x-3}\]