Distance between 2 stars is 10a. The masses of these stars are M and 16 M and their radii a and 2a. A body of mass m is fired straight from larger star towards smaller star. What should be its minimum initial speed to reach the surface of smaller star ?

Let P be the point on the line joining the centres of the two planets such that the net field at that point is zero.
Then, r2GM​−(10a−r)2G16M​=0
⇒(10a−r)2=16r2
⇒10a−r=4r
⇒r=2a
Potential at point P,
vp​=r−GM​−(10a−r)G16M​=2a−GM​−a2GM​
=2a−5GM​
Now if the particle projected from the larger planet has enough energy to cross this point, it will reach the smaller planet.
For this, the KE imparted to the body must be just enough to raise its total mechanical energy to a value which is equal to PE at point P, i.e.,
21​mv2−2aG(16M)m​−8aGMm​=mvp​
or 2v2​−a8GM​−8aGM​=−2a5GMm​
or v2=4a45GM​
or vmin​=23​a5GM​​.