Please see attachments for question.
Please use i, j and k vector coordinates in solving if possible.
DC = sqrt(L^2 + r^2)
Angle ADC = phi
sin(phi) = AC/DC = r/sqrt(L^2 + r^2)
cos(phi) = AD/DC = L/sqrt(L^2 + r^2)
Angle CDB = theta - phi
Radius of axle BC = (R = BC) = CD*sin(theta-phi)
=> R = sqrt(L^2 + r^2)*[sin(theta)cos(phi) - cos(theta).sin(phi)]
=> R = sqrt(L^2 + r^2)*[L*sin(theta) - r*cos(theta)]/sqrt(L^2 + r^2)
=> R = ...
The solution provides a clear, step-by-step solution to the question attached on general motion in physics.