Find the equation of all lines having slope ` 1`that are tangents to the curve `y=1/(x-1), x!=1`.

Answer

The equations of the lines are

`y+x+1=0` and `y+x-3=0`

Explanation

Given the equation of curve is `y=\frac{1}{x-1}, x \ne 1`.


Also ,the slope of the tangents to curve at any point `(x, y)` is;
$$ \frac{d y}{d x}=\frac{-1}{(x-1)^{2}} $$
If the slope of the tangent is `(-1)`, then we have:
$$
\begin{aligned}
...