find equations of the tangent lines to the curve y= (x-1)/(x+1) that are parallel to the line x-2y=2

Question

find equations of the tangent lines to the curve 

 

y= (x-1)/(x+1)

 

that are parallel to the line x-2y=2

Answer 1

The derivative of y is going to be equal to the slope of the given line

 

-2y = -x + 2

 

   y = (1/2)x - 1

 

 

Now that we know the slope of the tangent line, we find the derivative of the rational function.

 

y' = [(x + 1) - (x - 1)] / (x + 1)²

 

y' = 2 / (x + 1)²

 

 

Set this equal to 1/2.

 

2 / (x + 1)² = 1 / 2

 

 

Cross-multiply and solve for x.  This will give you the points of tangency.

 

4 = (x + 1)²

 

±2 = x + 1

 

-1 ± 2 = x

 

x = -3         and        x = 1

 

 

The tangent lines parallel to the given line occur at x=-3 and x=1.  Evaluate the rational function at these x values.  You will get the two points (-3, 2) and (1, 0).

 

Finally, get your two lines using the point-slope form.

 

Your first line is

 

y = (1/2)x - (1/2)(-3) + 2

y = (1/2)x + (3/2) + 2

y = (1/2)x + (7/2)

 

 

Your second line:

 

y = (1/2)x - (1/2)(1) + 0

y = (1/2)x - (1/2)