Answer to Question #125354 in Analytic Geometry for Samuel kassapa

Question

Find the points on the graph of the function f(x)=x+1/x-1. Where the slope of the tangent line is equal to -1/3 determine the equation of the tangent and normal to line

Answer 1

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

a formula of the derivative f(x)=1*(x-1)-(x+1)*1/(x-1)²=-2/(x-1)²

the derivative of the function is equal to the tangent of the angle of inclination

f(x₀)=-1/3=-2/(x₀-1)²,

1/6=(x₀-1)²,

x₀₁-1=√6 , x₀₁=1+√6,

x₀₂-1=-√6, x₀₂=1-√6,

F(x₀₁)=(2+√6)/(√6) =1+√2/√3,

F(x₀₂)=(2-√6)/(-√6)=1-√2/√3.

Equations of the tangent y=f(x₀)(x-x₀)+F(x₀),

y₁=-1/3(x-1-√6)+1+√2/√3,

y2=-1/3(x-1+√6)+1-√2/√3.

Equations of the normal y=-1/f(x₀)(x-x₀)+F(x₀),

y1=3(x-1-√6)+1+√2/√3,

y2=3(x-1+√6)+1-√2/√3.