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