Orthogonal Trajectories

Calculus 2 · Axiom Academy

LESSON Orthogonal Trajectories Every family of curves has a partner family that crosses it at right angles — and one slope rule finds it. 1. Perpendicular Means Negative Reciprocal Two curves are orthogonal at a point when their tangent lines meet at a right angle there. From slopes alone, that says everything: if the given curve has slope m , the curve crossing it perpendicularly must have slope — the negative reciprocal . Their slopes multiply to -1 . 2. The Method: Flip the Slope, Solve Again The given family hides a parameter (the c that names which curve you're on). The recipe clears it, flips the slope, and re-solves: Differentiate the family and use the original equation to eliminate the parameter, leaving a pure slope . Replace that slope with its negative reciprocal: . This is the orthogonal slope at every point. Integrate the new differential equation. Its general solution is the family of orthogonal trajectories. Flipping rotates the entire slope field by 90° — so the curves that thread it turn perpendicular. Take the circles through the origin with centers on the x -axis: x^2 + y^2 = 2cx . Run the method once and a twin family falls out — circles through the origin with centers on the y -axis.

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