Families of curves orthogonal to the lines $y=mx-2m-m^3$

TL;DR

This paper explores the family of curves orthogonal to a known line family, revealing new 'parabola-like' and non-parabola-like curves by solving a complex differential equation.

Contribution

It introduces new families of curves orthogonal to a given line family, extending the understanding of orthogonal trajectories beyond the classical parabola.

Findings

Derived new orthogonal trajectories to the line family.

Identified 'parabola-like' and non-parabola-like curves.

Solved a complex first and third degree differential equation.

Abstract

The family of lines $y=mx-2m-m^3$, are well known to be normal to the parabola $y^2=4x$. However, this family of lines is normal to a family of curves of which this parabola is just one member. Here, by solving an interesting first order and third degree ODE, we bring out these curves. The resulting one set of curves are "parabola-like" but non-standard ones and the other family is not even "parabola like".