# A tangent line is drawn to the hyperbola #xy=c# at a point P, how do you show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P?

##### 1 Answer

We have

Let us suppose that P has x-coordinates

So the gradient of the tangent at P is given by

when

The tangent passes through

Now Let's find the midpoint of the tangent line as its passes through the axis.

The tangent cuts the

The tangent cuts the

This yields two solutions:

1) Either,

#t=0# which corresponds to a vertical asymptote#(0,oo)# which we can dismiss as a valid solution

2) Or#ty-2c=0 => y=(2c)/t#

So the tangent touches the axis at

So the mid-pint of these two coordinates is: