M = mass of earth

M₁ = mass of shaded portion

R_e = Radius of earth

M₁ = $${M \over {{4 \over 3}\pi {R^3}}}.{4 \over 3}\pi {\left( {R - h} \right)^3}$$

= $${{M{{\left( {R - h} \right)}^3}} \over R^3}$$

Given, Weight of body is same at P and Q

$$ \therefore $$ mg_P = mg_Q

$$ \Rightarrow $$ g_P = g_Q

$$ \Rightarrow $$ $${{G{M_1}} \over {{{\left( {R - h} \right)}^2}}} = {{GM} \over {{{\left( {R + h} \right)}^2}}}$$

$$ \Rightarrow $$ $${{GM{{\left( {R - h} \right)}^3}} \over {{R^3}{{\left( {R - h} \right)}^2}}} = {{GM} \over {{{\left( {R + h} \right)}^2}}}$$

$$ \Rightarrow $$ (R – h) (R + h)² = R³

$$ \Rightarrow $$ R³ – hR² + h²R – h³ + 2R²h – 2Rh² = R³

$$ \Rightarrow $$ R²h – Rh² – h³ = 0

$$ \Rightarrow $$ R² – Rh – h² = 0

$$ \Rightarrow $$ h² + Rh – R² = 0

$$ \Rightarrow $$ h = $${{ - R \pm \sqrt {{R^2} + 4{R^2}} } \over 2}$$

$$ \Rightarrow $$ h = $${{ - R + \sqrt 5 R} \over 2}$$