At what distance from the centre of the Moon is the point at which the strength of the resultant of the Earth's and Moon's gravitational fields is equal to zero? The Earth's mass is assumed to be `eta=81` times that of the Moon, and the distance between the centres of these planets `n=60` times greater than the radius of the Earth R.

Explanation

Let r be the sought distance, then
`(gammaetaM)/((nR-r)^2)=(gammaM)/(r^2)` or `etar^2=(nR-r)^2`
or `sqrtetar=(nR-r)` or `r=(nR)/(sqrt(eta+1))=3*8xx10^4km`.