Q.A point in the plane moves so that its distance from the origin is thrice its distance from the $y$-axis. Find its locus.

A locus is the algebraic translation of a geometric rule that a moving point obeys. The working method is always the same: name the moving point $P(x,y)$, state its distance condition(s) using the distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, square to clear the root, and simplify. A condition of "equidistant from two fixed points" always collapses to a first-degree equation — a straight line, specifically the perpendicular bisector of the two points. A condition of "equidistant from a point and a line" instead leaves one variable squared — a parabola. A c …