Find an equation?

Question

Find an equation of a curve that intersects at right angles every curve of the family y=1/x+k (where k takes all real values)?

 

Answer: y=(1/3)x^3

 

 

Answer 1

 f(x) = 1/(x + k)

 f'(x)  = -1/ (X + k)²      g(x) =the function which is perpendicular to f(x)

 

 

    g^' ( X) = -1/ f'(x)  = -1 / ( -1/ x +k)²= (X +k) ²

 

     g(x) = ∫ ( X + k) ^2) dx =  1/3 ( X + k ) ³ + C