JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Straight Lines

Since $P(5,4)$ and $Q(-2,4)$ share the same $y$-coordinate, $PQ$ is horizontal with length $7$, and the altitude from $R$ to $PQ$ is vertical. So the orthocenter's $x$-coordinate equals $R$'s $x$-coordinate: $a=2$.



To fix the sign, use that $\overrightarrow{PO}$ must be perpendicular to $\overrightarrow{QR}$ (the altitude from $P$):



So $R=(2,-6)$. The centroid is