If the distance between centres of earth and moon is D and the mass of earth is 81 times the mass of moon, then at what distance from centre of earth the gravitational force will be zero

The correct option is 9D/10.

Step-by-step Derivation:

Let us define the variables given in the problem:
- The distance between the centres of the Earth and the Moon is $D$.
- The mass of the Earth is $M_E$.
- The mass of the Moon is $M_M$.
According to the question, the mass of the Earth is 81 times the mass of the Moon:

$M_E=81M_M$

Let there be a point between the Earth and the Moon at a distance $x$ from the centre of the Earth where the net gravitational force on a test mass $m$ is zero.
Consequently, the distance of this point from the centre of the Moon will be:

$D-x$

At this neutral point, the gravitational pull exerted by the Earth on the test mass must be equal in magnitude and opposite in direction to the gravitational pull exerted by the Moon. Therefore, we equate the two forces:

$\frac{G{M}_{E}m}{x^2}=\frac{G{M}_{M}m}{{(D-x)}^2}$

Here, $G$ is the universal gravitational constant. We can cancel $G$ and the test mass $m$ from both sides of the equation:

$\frac{M_E}{x^2}=\frac{M_M}{{(D-x)}^2}$

Now, substitute the relation $M_E=81M_M$ into the equation:

$\frac{81M_M}{x^2}=\frac{M_M}{{(D-x)}^2}$

Divide both sides by $M_M$:

$\frac{81}{x^2}=\frac{1}{{(D-x)}^2}$

Taking the square root on both sides of the equation to simplify:

$\frac{9}{x}=\frac{1}{D-x}$

Cross-multiplying the terms gives:

$9(D-x)=x$

Expand the left side of the equation:

$9D-9x=x$

Rearrange the terms to solve for $x$:

$10x=9D$

$x=\frac{9D}{10}$

Thus, the gravitational force will be zero at a distance of 9D/10 from the centre of the Earth.