Angle between orthogonal trajectory and normals of a given family of curves (in degrees) at a given point on the curve = ___

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Solution

To answer this question, we should know what orthogonal trajectory is. Orthogonal trajectory is defined for a family of curves. General solutions we get after solving differential equations are family of curves. If there is an arbitrary constant in the equation of a curve, it basically represents a family of curves. Depending on the value of the constant, we get different curves. y = x+c represents family of straight lines whose slope is $45^{\circ}$ . Family of curves which are orthogonal or perpendicular to a given family of curves is defined as the orthogonal trajectory of those given family of curves. Angle between orthogonal trajectory and the given set of curves will be $90^{\circ}$ . We know that angle between normal and a curve is also $90^{\circ}$ . So the angle between normal and orthogonal trajectory will be zero because both are perpendicular to the curve.

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