Tangent Line Problem?

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 tangent line of  the curve y = (x - 1) / (x + 1)

Differenciate each side with respective x

                                                    dy / dx  = (x + 1)(1 - 0) - (x - 1)(1 + 0) / (x + 1)²

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

                                                               m  = 2 / (x + 1)²

The line equation : x - 2y = 2

The slope of the line : m = - xcoefficient / y coefficient = -1 / -2 = 1 / 2.

Two lines are parallel then the slopes are equal

Therefore m = 2 / (x + 2)² = 1 / 2

                          4 = (x + 2)²

Let x = 0 and -4 then 4 = (0 + 2)²

                         4 = 2² = 4

Substitute x = 0 in the tangent in the tangent line of the curve

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

                             y  = -1 / 1 = -1

y = -4 - 1 / -4 + 1 = -5 / -3 = 5 / 3

The point is (0, -1) and (-4 , 5 / 3)

The tangent line equation form y - y1 = m (x - x1)

The point (0,-1) and slope m = 1 / 2 in the tangent line equation

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

y + 1 = x / 2

2y + 2 = x

x - 2y = 2

The point(-4,5 / 3) in the tangent line equation

y + 4 = 1 / 2(x - 5 / 3)

2y + 8 = (x - 5 / 3)

x - 2y = 8 + 5 / 3 = 29 / 3.

The tangent line equation : x - 2y = 29 / 3.

Comments

The tangent lines are y = x/2 - 1/2 and y = x/2 + 7/2.

Answer 2

The curve equation is y = (x - 1)/(x + 1) and parallel line x - 2y = 2.