Find all points on the graph of y=(x+1)/(x) where the tangent line is parallel to the line x+9y=4.

Answer 1

Slope of the tangent line to the graph y = (x+1) / x at point (a,b) is

m₁ = [x - (x+1)] / (x²) at (a,b)

m₁ = -1/a²

Slope of the given line is m₂ = -1/9

m₁ = m₂

⇒ a² = 9

⇒ a = ± 3

(a,b) is also a point on the graph y = (x+1) / x itself. Therefore, b = (a+1) / a

For a = 3, we get b = 4/3

For a = -3, we get b = 2/3

Therefore, there are 2 such points: (3, 4/3) and (-3, 2/3)

Answer 2

y' = - 1/x² = - 1/9, 1/x² = 1/9, x² = 9, x = ± 3.

If x = 3 then y = 4/3; point (3, 4/3)

If x = - 3 then y = 2/3, point (- 3, 2/3)