Final Answer:
The orthogonal trajectories of the family of curves y = kx⁷ are given by the equation x = ky⁷.
Step-by-step explanation:
Orthogonality definition:
Orthogonal trajectories are families of curves that intersect each other at right angles. This means that their slopes are negative reciprocals of each other.
Differentiate the original equation:
dy/dx = 7kx⁶
Negative reciprocal slope:
-1/(dy/dx) = -1/(7kx⁶) = -kx⁻⁶
Replace y with x and x with y:
x = ky⁻⁶
Rearrange the equation:
x = ky⁷
Therefore, the orthogonal trajectories of the family of curves y = kx⁷ are given by the equation x = ky⁷.