If $\mathbf{a},{\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3$…….. and  $\mathbf{b},{\mathbf{y}}_1,{\mathbf{y}}_2$ ….. form two infinite G.P’s with same common ratio ‘r’ then the locus of P(h, k) when $\mathbf{h}\mathbf{=}\sqrt[n]{{\mathbf{x}}_1{\mathbf{x}}_2{\mathbf{x}}_3\mathrm{......}{\mathbf{x}}_n}$ and  $\mathbf{k}\mathbf{=}\sqrt[n]{{\mathbf{y}}_1{\mathbf{y}}_2{\mathbf{y}}_3\mathrm{......}{\mathbf{y}}_n}$ is

$\begin{array}{l}{\text{x}}_{\text{1}}{\text{=ar \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y}}_{\text{1}}\text{=\hspace{0.17em}\hspace{0.17em}br}\\ {\text{x}}_{\text{2}}{\text{\hspace{0.17em}=\hspace{0.17em}ar}}^{\text{2}}{\text{\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y}}_{\text{2}}{\text{\hspace{0.17em}=\hspace{0.17em}br}}^{\text{2}}\end{array}$

$\begin{array}{c}•\\ •\\ •\end{array}$              $\begin{array}{c}•\\ •\\ •\end{array}$

${\text{x}}_{\text{n}}{\text{=\hspace{0.17em}ar}}^{\text{n-1}}{\text{\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y}}_{\text{n}}{\text{=\hspace{0.17em}br}}^{\text{n-1}}$

$\text{h\hspace{0.17em}=\hspace{0.17em}}\sqrt[\text{n}]{{\text{x}}_{\text{1}}{\text{\hspace{0.17em}x}}_{\text{2}}\mathrm{.........}{\text{x}}_{\text{n}}}{\text{\hspace{0.17em}=a.\hspace{0.17em}r}}^{\frac{\text{n-1}}{\text{2}}}$

Similarly  ${\text{k=b.r}}^{\frac{\text{n-1}}{\text{2}}}\text{;\hspace{0.17em}\hspace{0.17em}}\frac{\text{h}}{\text{k}}\text{=}\frac{\text{a}}{\text{b}}\Rightarrow \text{\hspace{0.17em}bh=ak}$
Locus is bx – ay = 0