Find equations of the tangent lines to the curve y = (x - 1)/(x + 1) that are parallel to the line x - 2y = 4

Solution:

The equation given is y = (x - 1)/(x + 1)

The derivative of the function y is

\(\frac{dy}{dx}=\frac{(x+1)\times 1- (x-1)\times 1}{(x+1)^{2}}=\frac{2}{(x+1)^{2}}\)

It is given that the equation is parallel to x - 2y = 4

We know that the equation of a line is y = mx + b

Where m is the slope, b is the y intercept

So the slope is y = 1/2x - 4/2

y = 1/2 x - 2

m = 1/2

So the derivative of the function y is 1/2

2/(x + 1)² = 1/2

(x + 1)² = 4

x² + 2x - 3 = 0

Splitting the middle term

(x + 3)(x - 1) = 0

x = - 3 and x = 1

If x = -3

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

So the point is (-3, 2)

If x = 1

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

So the point is (1, 0)

The point slope equation is

y - y₁ = m (x - x₁)

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

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

y - 2 = x/2 + 3/2

y = 1/2 x + 7/2

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

y = 1/2 x - 1/2

Therefore, the equations of the tangent lines are y = 1/2 x + 7/2 and y = 1/2 x - 1/2.

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Find equations of the tangent lines to the curve y = (x - 1)/(x + 1) that are parallel to the line x - 2y = 4

Summary:

The equations of the tangent lines to the curve y = (x - 1)/(x + 1) that are parallel to the line x - 2y = 4 are y = 1/2 x + 7/2 and y = 1/2 x - 1/2.