Final answer:
The orthogonal trajectories for y = kx⁴ are parabolas.
Step-by-step explanation:
An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci is a constant. It can be formed by the intersection of a plane with a cone. The equation for an ellipse can be written as x = ay² + by + c, where a, b, and c are constants.
In the given equation y = kx⁴, we can rewrite it as x⁴ = (1/k)y. Comparing this with the general form for an ellipse equation, we can see that it matches the form x = ay² + by + c. Therefore, the orthogonal trajectories for y = kx⁴ are parabolas.