Locus of Midpoint of tangent to Hypernola

Suppose a tangent is drawn to an hyperbola. The hyperbola intersects the two axes to form a line of finite length. What is the equation of the midpoint of this line?
A general point on the hyperbola is given in parametric coordinates as  
\[(acosht, b sinht)\]
  and the gradient of the hyperbola at this point is  
\[\frac{dy}{dx}= \frac{dy/dt}{dx/dt}= \frac{bcosht}{asinht}\]
.
The  
\[y\]
  intercept - using  
\[y=mx+c\]
  - is the solution  
\[c\]
  to  
\[bsinht= \frac{bcoshtt}{asinht} \times acosht+c \rightarrow c=bsinht-\frac{bcosht}{asinht} \times acohst=\frac{bsinh^2t-bcosh^2t}{sinht}=-\frac{b}{sinht}\]

The equation of the tangent is  
\[y= \frac{bcosht}{asinht}x-\frac{b}{sinht}\]
.
The  
\[x\]
  intercept is the solution to  
\[0= \frac{bcosht}{asinht}c-\frac{b}{sinht} \rightarrow x= \frac{b/sinht}{bcosht/asinht}=\frac{a}{cosht}\]

The  
\[x\]
  intercept is  
\[\frac{b}{sinht}\]
.
The coordinates of the intercepts are  
\[(\frac{a}{cosht},0)\]
  and  
\[(0,\frac{b}{sinht})\]
  and the coordinates of the midpoints are  
\[(\frac{a}{2cosht}, \frac{b}{2sinht} )\]
.
We get  
\[x=\frac{a}{2cosht} \rightarrow cosht=\frac{a}{2x}\]
  and  
\[y=\frac{b}{2sinht} \rightarrow sint= \frac{b}{2y}\]
.
Then  
\[cosh^2-sinh^2t=1=\frac{a^2}{4x^2}-\frac{b^2}{4y^2}\]
.