JEE Math Practice Question

Consider the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ defined by $\mathrm{L}_1: \mathrm{x} \sqrt{2}+\mathrm{y}-1=0$ and $\mathrm{L}_2: \mathrm{x} \sqrt{2}-\mathrm{y}+1=0$ For a fixed constant $\lambda$, let $\mathrm{C}$ be the locus of a point $\mathrm{P}$ such that the product of the distance of $\mathrm{P}$ from $\mathrm{L}_1$ and the distance of $\mathrm{P}$ from $\mathrm{L}_2$ is $\lambda^2$. The line $\mathrm{y}=2 \mathrm{x}+1$ meets $\mathrm{C}$ at two points $\mathrm{R}$ and $\mathrm{S}$, where the distance between $\mathrm{R}$ and $\mathrm{S}$ is $\sqrt{270}$. Let the perpendicular bisector of RS meet $\mathrm{C}$ at two distinct points $\mathrm{R}^{\prime}$ and $\mathrm{S}^{\prime}$. Let $\mathrm{D}$ be the square of the distance between $\mathrm{R}^{\prime}$ and S'. What is the value of $\lambda^{2}$?