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

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

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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

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