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(D) Let the mass of the Moon be $M$ and the mass of the Earth be $81M$.Let the distance from the centre of the Earth where the gravitational field is

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$\frac{9D}{10}$

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(D) Let the mass of the Moon be $M$ and the mass of the Earth be $81M$.Let the distance from the centre of the Earth where the gravitational field is zero be $x$.Then the distance from the centre of the Moon is $(D - x)$.The gravitational field due to the Earth at this point is $E_E = \frac{G(81M)}{x^2}$.The gravitational field due to the Moon at this point is $E_M = \frac{GM}{(D - x)^2}$.For the net gravitational field to be zero,$E_E = E_M$.$\frac{G(81M)}{x^2} = \frac{GM}{(D - x)^2}$.$\frac{81}{x^2} = \frac{1}{(D - x)^2}$.Taking the square root on both sides,we get $\frac{9}{x} = \frac{1}{D - x}$.$9(D - x) = x$.$9D - 9x = x$.$9D = 10x$.$x = \frac{9D}{10}$.

If the distance between the centres of Earth and Moon is $D$ and the mass of Earth is $81$ times that of the Moon,at what distance from the centre of Earth will the gravitational field be zero?

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