Question

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

Answer 1

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.