Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 2 / v / Solution

At a Triangular Lagrange point, $y\ne 0$, so $A=1$. The $x$ equation then givesSubstitution into $A=1$ and ${\mu }_1+{\mu }_2=1$ yieldsIntersecting these two circles givesAs ${\beta }_1$ rises from zero to one, $s$ falls from one to zero. The two points move along the unit circle about $M_2$, from the classical equilateral positions toward $M_1$, where they coalesce when the attraction of $M_1$ is fully cancelled.