Answer to Question #229221 in Chemical Engineering for Lokika

Question

Find the equation of the family of all erhogonal trajectories of the family ofcircles which pass through origin and hav centres on the y-axis.

Answer 1

The circle through the origin having center (0,a)(0,a) has equation x2+(y−a)2=a2x2+(y−a)2=a2, and we seek curves y=y(x)y=y(x) with the property that if it passes through a point (x0,y0)(x0,y0) on this circle then it is orthogonal to the circle at this point. The slope required for this is the slope of the radius to the circle at this point which is y0−ax0y0−ax0, the slope of the segment from (0,a)(0,a) to (x0,y0)(x0,y0).