2021 Fall AMC 10B Problem 17

Distinct lines \ell and m lie in the xy-plane. They intersect at the origin. Point P(-1, 4) is reflected about line \ell to point P', and then P' is reflected about line m to point P''. The equation of line \ell is 5x - y = 0, and the coordinates of P'' are (4,1). What is the equation of line m?

(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0 (\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.