JEE Main · 2025 · 07 Apr 2025 · Shift 2 · Straight Lines

The vertices on $y=x+1$ and $y=4x-8$ meet at $P(3,4)$, and the orthocentre is $H(3,-1)$. Since $PH$ is the vertical line $x=3$, the altitude from $P$ is vertical, so the opposite side $y=mx+c$ must be horizontal, giving $m=0$. With the third side $y=c$, let $B=(c-1,c)$ be its intersection with $y=x+1$. The altitude from $B$ is perpendicular to $y=4x-8$, hence has slope $-\frac{1}{4}$ and passes through $H$: $\frac{-1-c}{3-(c-1)}=-\frac{1}{4}$ gives $c=0$. Therefore $m-c=0$.