JEE Main · 2026 · 05 Apr 2026 · Shift 2 · Straight Lines

The line $x+y=0$ has inclination $135^{\circ }$; rays at $45^{\circ }$ to it have inclinations $90^{\circ }$ and $180^{\circ }$, so through $(-1,-1)$ they are $x=-1$ and $y=-1$. Reflection in the mirror $x+2y-1=0$ sends $(x_0,y_0)$ to $\left(x_0-\frac{2(x_0+2y_0-1)}{5},y_0-\frac{4(x_0+2y_0-1)}{5}\right)$, so the common point $(-1,-1)$ maps to $\left(\frac{3}{5},\frac{11}{5}\right)$. The ray $x=-1$ strikes the mirror at $(-1,1)$, and the reflected ray through $(-1,1)$ and $\left(\frac{3}{5},\frac{11}{5}\right)$ has slope $\frac{3}{4}$, giving $-3x+4y=7$. The ray $y=-1$ strikes at $(3,-1)$, and the reflected ray through $(3,-1)$ and $\left(\frac{3}{5},\frac{11}{5}\right)$ has slope $-\frac{4}{3}$, giving $4x+3y=9$. Matching $ax+by=9$ and $cx+dy=7$ gives $a=4,b=3,c=-3,d=4$, so $ad+bc=16-9=7$.