If $$R_E$$ be the radius of Earth, then the ratio between the acceleration due to gravity at a depth $$r$$ below and a height $$r$$ above the earth surface is: (Given: $$r \lt R_E$$)

We have to find the ratio between the acceleration due to gravity at a depth $$r$$ below the Earth’s surface and at a height $$r$$ above the surface, where $$r \lt R_E$$ and $$R_E$$ is the Earth’s radius.

First recall the standard results for the variation of acceleration due to gravity with depth and with height:

1. Variation with depth. For a depth $$r$$ (or $$d$$) inside the Earth, the formula is stated as $$g_d = g\left(1 - \frac{r}{R_E}\right)$$ because only the mass enclosed within radius $$R_E - r$$ contributes to the gravitational pull, leading to a linear decrease.

2. Variation with height. For a height $$r$$ (or $$h$$) above the surface, the formula is stated as $$g_h = \frac{g}{\left(1 + \frac{r}{R_E}\right)^2}$$ because the field outside the Earth follows the inverse-square law while the distance from the centre becomes $$R_E + r$$.

Now we are asked for the ratio $$\dfrac{g_d}{g_h}$$. Substituting the two formulae, we write

$$ \frac{g_d}{g_h} = \frac{g\left(1 - \frac{r}{R_E}\right)} {\displaystyle \frac{g}{\left(1 + \frac{r}{R_E}\right)^2}} = \left(1 - \frac{r}{R_E}\right)\left(1 + \frac{r}{R_E}\right)^2 . $$

The factor $$g$$ cancels out, as expected. To make the algebra clearer, set

$$x = \frac{r}{R_E}, \qquad \text{with } x \lt 1.$$

With this substitution the ratio becomes

$$ \frac{g_d}{g_h} = (1 - x)(1 + x)^2 . $$

Now expand every factor step by step. First expand the square:

$$ (1 + x)^2 = 1 + 2x + x^2 . $$

Next multiply this result by the remaining factor $$(1 - x)$$:

$$ (1 - x)(1 + 2x + x^2) = 1 + 2x + x^2 \;-\; x - 2x^2 - x^3 . $$

Combine like terms:

$$ 1 + 2x + x^2 - x - 2x^2 - x^3 = 1 + (2x - x) + (x^2 - 2x^2) - x^3 = 1 + x - x^2 - x^3 . $$

Finally replace $$x$$ by $$\dfrac{r}{R_E}$$ to return to the original variables:

$$ \frac{g_d}{g_h} = 1 + \frac{r}{R_E} - \frac{r^2}{R_E^2} - \frac{r^3}{R_E^3}. $$

This result matches exactly the expression given in Option A.

Hence, the correct answer is Option A.