\(M _{1}= mass\) of shaded portion

\(R =\) Radius of earth

\(M_{1}=\frac{M}{\frac{4}{3} \pi R^{3}} \cdot \frac{4}{3} \pi(R-h)^{3}\)

\(=\frac{M(R-h)^{3}}{R}\)

Weight of body is same at \(P\) and \(Q\)

i.e. \(mg _{ P }= mg _{ Q }\)

\(g _{ P }= g _{ Q }\)

\(\frac{G M_{1}}{(R-h)^{2}}=\frac{G M}{(R+h)^{2}}\)

\(\frac{G M(R-h)^{3}}{(R-h)^{2} R^{3}}=\frac{G M}{(R+h)^{2}}\)

\(( R - h )( R + h )^{2}= R ^{3}\)

\(R^{3}-h R^{2}-h^{2} R-h^{3}+2 R^{2} h-2 R h^{2}=R^{3}\)

\(R^{2}-R h^{2}-h^{3}=0\)

\(R^{2}-R h-h^{2}=0\)

\(h ^{2}+ Rh - R ^{2}=0 \Rightarrow h =\frac{- R \pm \sqrt{ R ^{2}+4 R ^{2}}}{2}\)

ie \(h =\frac{- R +\sqrt{5} R }{2}=\left(\frac{\sqrt{5}-1}{2}\right) R\)