1.

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

Answer» Equation of the curve,
`y = 1/(x-1)`
`:.` Slope of tangent,`( dy/dx) = -1/(x-1)^2`
Slope of tangent to this curve is given `-1`.
`:. -1/(x-1)^2= -1`
`:. x = 0 and x = 2`
`=>y = -1 and y = 1`
Therefore, equation of tangents to the given curve,
`(y+1)/(x-0) = -1 and (y-1)/(x-2) = -1`
`=>x+y+1 = 0 and x+y-3 = 0`, which are the required equations.


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