JEE Main 2026 Apr 5 Shift 2, Mathematics Q22: Points and Straight Lines

JEE Main2026Apr 5, Shift 2Mathematics

From the point $(- 1, - 1)$ , two rays are sent making angles of $45°$ with the line $x + y = 0$ . These rays get reflected from the mirror $x + 2 y = 1$ . If the equations of the reflected rays are $a x + b y = 9$ and $c x + d y = 7$ , $a, b, c, d \in Z$ , then the value of $a d + b c$ is ______.

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Solution

The two rays from $(- 1, - 1)$ at $45°$ to $x + y = 0$ (slope $- 1$ ) have slopes $0$ and $\infty$ . So the incident rays are $L_{1} : x = - 1$ and $L_{2} : y = - 1$ .

The reflected ray is obtained by reflecting the incident line in the mirror $x + 2 y = 1$ .

Reflecting $L_{1} : x + 1 = 0$ in the mirror: the image of any point on $L_{1}$ across $x + 2 y = 1$ traces out the reflected line. Using the standard reflection formula, a representative point on $L_{1}$ reflects to a point that, with the fixed intersection on the mirror, gives the line $3 x - 4 y + 7 = 0$ , i.e. $- 3 x + 4 y = 7$ . Matching with $c x + d y = 7$ : $c = - 3$ , $d = 4$ .

Reflecting $L_{2} : y + 1 = 0$ : by the same procedure, the reflected line is $4 x + 3 y = 9$ . Matching with $a x + b y = 9$ : $a = 4$ , $b = 3$ .

$∴ a d + b c = 4 \cdot 4 + 3 \cdot (- 3) = 16 - 9 = 7$ .

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JEE Main 2026 Apr 5 Shift 2, Mathematics Q22: Points and Straight Lines

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