Calculus question about orthogonal trajectories

Question

the question states Find and describe the orthogonal trajectories of the family of curves y = ln(kx).

how would i solve this question

Answer 1

The slope of the curve y = ln( k x) is 1/x for any value of k.

The negative reciprocal of this is -x. The function with slope = -x is Y(x) = -0.5 x² +C

So the family of curves Y(x) = -0.5 x² +C is a set of trajectories orthogonal to ln( k x)