Question

Find the equation of all straight lines which are tangent to curve $y=\frac{1}{x-1}$ and which are parallel to line x + y =0

Answer 1

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

$\frac{dy}{dx}=\frac{\left(x-1\right)\times 0-\left(1-0\right)}{{\left(x-1\right)}^2}$

$=\frac{-1}{{\left(x-1\right)}^2}$

${\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}=-\frac{1}{{\left(x,-1\right)}^2}$

$\frac{-1}{{\left(x,-1\right)}^2}=-1$

${\left(x-1\right)}^2=1$

$x_1-1=\pm 1$

$x_1=0,2$

$y_1=-1,1$

$Equationis$

$y+1=-\left(x\right),y-1=-\left(x-2\right)$

$x+y+1=0and,$

$x+y-3=0$

$whicharetherequiredanswer.$