JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Straight Lines

The lines $4x+5y=0$ and $7x+2y=0$ meet at the origin, so the origin is a vertex of the parallelogram. The given diagonal $11x+7y=9$ does not pass through the origin, so it joins the two vertices adjacent to the origin.
Solving $11x+7y=9$ with $4x+5y=0$ gives $\frac{27x}{5}=9$, so one of those vertices is $\left(\frac{5}{3},-\frac{4}{3}\right)$. Solving it with $7x+2y=0$ gives $-\frac{27x}{2}=9$, so the other is $\left(-\frac{2}{3},\frac{7}{3}\right)$.
The vertex opposite the origin is the sum of these two position vectors:

The other diagonal joins $(0,0)$ to $(1,1)$, so it is the line $y=x$, which passes through $(2,2)$.