A family of curves is given by \(x^2+kx=ky-y^2\), where is \(k\) is an arbitrary constant. Find a set of orthogonal trajectories to this family.

Details: A set of orthogonal trajectories is comprised of a family of curves whose tangents at any point of intersection with the original curves are at right angles (perpendicular) with the tangents to the original curves. In the choices below \(c\) is an arbitrary constant.

The choices are:

A) \(\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{c^2}\)

B) \(y^2-x^2=c^2\)

C) \((y+c)^2-(x-c)^2=1\)

D) \(y=\frac{c}{x}\)

E) \(x^2-3y^2=c\)

F) \((x-c)^2+(y-c)^2=2c^2\)

G) \(\frac{y^2}{2}=x^2+c\)

The image above obviously represents different families of curves from those described by the equation in this problem.

Inspiration: Hosam Hajjir: Orthogonal Trajectories - Part 2

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