JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Straight Lines

Write the line in intercept form $\frac{x}{p}+\frac{y}{q}=1$, where $p$ and $q$ are its intercepts on the axes. The mean of the reciprocals of the intercepts being $\frac{1}{4}$ means $\frac{1}{2}\left(\frac{1}{p}+\frac{1}{q}\right)=\frac{1}{4}$, so $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$. Evaluating the intercept equation at $(2,2)$ gives $\frac{2}{p}+\frac{2}{q}=2\left(\frac{1}{p}+\frac{1}{q}\right)=1$, so $(2,2)$ satisfies the line for every such choice of intercepts. The path therefore always passes through $(2,2)$, the position of stone $B$, while $(1,1)$ and $(4,4)$ carry no such guarantee.