A large spherical mass M is fixed at one position and two identical...

Due to gravitational interaction connected masses have some acceleration.
Let both small masses are moving with acceleration 'a' towards larger mass M


Force equation for mass nearer to larger mass
$\dfrac{GMm}{{\left(3l\right)}^{2}} - \dfrac{G{m}^{2}}{{l}^{2}} = ma$ ...(i)
Force equation for mass away from larger mass
$\dfrac{GMm}{{\left(4l\right)}^{2}} + \dfrac{G{m}^{2}}{{l}^{2}} = ma$ ...(ii)
From equation (i) & (ii)
$\dfrac{GM}{9{l}^{2}} - \dfrac{Gm}{{l}^{2}} = \dfrac{GM}{16{l}^{2}} + \dfrac{Gm}{{l}^{2}}$
$⇒ \dfrac{M}{9} - \dfrac{M}{16} = m + m$
$\dfrac{7M}{144} = 2m$
$⇒ m = \dfrac{7M}{288} = k$\left(\dfrac{M}{288}\right)$<br >$⇒ K = 7