Find the equation of all straight lines which are tangent to curve `y=(1)/(x-1)` and which are parallel to the line x+y =0.

Explanation

Suppose the tangent is at `(x,. Y,)` and it has slope -1

`rArr" " (dy)/(dx)|_((x_(1) ,y)) =-1`
`rArr" " -(1)/((x_(1) -1)^(2) =-1`
`rArr " " x_(1) =0" "or " "2`
`rArr " " y_(1)=1 " or " "1`
Hence tangent at `(0,-1) " and " (2,1)` and the required lines (see figure ) with equations
`-1(X-0)= (y+1) " and " -1(x-2)=(y-1)`
`rArr " " X+y +1=0 " and " y+ x=3`