If a particle of mass z is projected with minimum velocity from the surface of a star with kinetic energy ${\mathrm{K}}_1\mathrm{GMm}/\mathrm{a}$ and potential energy at surface of the star ${\mathrm{K}}_2\mathrm{GMm}/\mathrm{a}$ towards the star of same mass m and radius a (K₁ and K₂ are constants) to reach the other star. Find the distance between the centre of the two stars

Since the mass and radius of two stars are same, point P at which gravitational force will be equal, will be at equal distance (say r) from centre of each star. The particle should be projected to cross P, beyond which the particle will be attracted towards the other star.

KE = PE of body at P- P at surface of the star

$\frac{{\mathrm{K}}_1\mathrm{GMm}}{\mathrm{a}}=\left[-\frac{\mathrm{GMm}}{\mathrm{r}}-\frac{\mathrm{GMm}}{\mathrm{r}}\right]-\left[-{\mathrm{K}}_2\frac{\mathrm{GMm}}{\mathrm{a}}\right]$

on solving, $$\begin{gathered} \mathrm{r}=\frac{2\mathrm{a}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}2 \\ \mathrm{r}=\frac{4\mathrm{a}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)} \end{gathered}$$