This is a University level homework question that I am having some difficulty with....

Find the equation of the line with the slope -1 that is the tangent to the curve

$\displaystyle

y=\frac{1}{x-1}

$

This is what I believe I should do:

use the slope to get the equation

$\displaystyle

y=-1x+k

$

Then set the y-values to equal eachother

$\displaystyle

\frac{1}{x-1}=-1x+k

$

set it to equal 0:

$\displaystyle

-1x^{2}+kx-k=0

$

Am I correct so far and where do I go from here?

Thank You