Find the equation of all lines having slope –1 that are tangents to the curve
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Step-by-step explanation:
The equation of the given curve is y=
1/x-1,x ≠1
The slope of the tangents to the given curve at any point (x,y) is given by,
dx/dy=-1/(x-1)^2
If the slope of the tangent is −1, then we have:
-1/(x-1)^2=-1
(x-1)^2=1
x-1=±1
x=2,0
When x=0, y=−1 and when x=2,y=1.
Thus, there are two tangents to the given curve with the slope −1. These are passing through the points (0,−1) and (2,1).
∴ The equation of the tangent through (0,−1) is given by,
y−(−1)=−1(x−0)⇒x+y+1=0
And equation of the tangent through (2,1) is given by,
y−1=−1(x−2)
⇒y+x−3=0
Hence, the equations of the required lines are y+x+1=0 and y+x−3=0
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