### Solution:

Concept- To calculate the distance to the point where gravitational fields are zero, we consider the distance from the moon to this point to be some variable 'x,' and we equalise the gravitational strengths of the earth and moon at this point to establish a relationship, then we solve this equation to find 'x.'Assume the distance between the cores of the Earth and the Moon is 'd.' Along the line of their radii, the earth gravitates towards itself and the moon gravitates towards itself. The gravitational force exerted by the earth equals that of the moon at a point 'P' where the sum of their gravitational forces equals zero. Let this point be y distance from the moon.

At point P, the earth's gravitational force equals that of the moon.

From the given,

Mass of earth Distance between earth and the point is " " and distance between moon and point is " ".

Distance between earth and moon Given that the distance between the earth's and moon's centres is 60 times the radius of the earth ' '.

Hence, The distance from the moon to the point where the net gravitational force is zero is six times the radius of the Earth, i.e.We know the radius of earth is 6371km. We know that the radius of the Earth is.

As a result, the distance from the Moon's core is the point at which the intensity of the resultant of the Earth's and Moon's gravitational forces equals zero.

Hence, the 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 is

$\n \n \n \n y=6\xd76371km\n \n \n \n \n =38226km\n \n \n \n \n =3.8\xd7\n 10\n 4\n \n km\n \n \n \n \n $ Hence, option 1 is the correct answer.