# Interacting Families

Geometry Level pending

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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