# Orthogonal Trajectories - Part 2

Geometry Level 5

A family of curves is given by $$y = c x^n$$, where $$n$$ is a fixed nonzero real number, and $$c$$ is an arbitrary constant. A set of orthogonal trajectories is comprised of a family of curves whose tangents at any point of intersection with the original curve are at right angles (perpendicular) with the tangents to the original curve. Which of the following choices represents the orthogonal trajectories to the above family? ($$k$$ is an arbitrary constant).

(A) $$y = \dfrac{k}{ x^n}$$

(B) $$y = k x^{\frac{1}{n}}$$

(C) $$y = k x^{1 - n }$$

(D) $$\dfrac{x^2}{n} + \dfrac{y^2}{1} = k^2$$

×