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Maths Question 11 – JEE-MAIN 2025

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Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle α with the positive x-axis and the equations of its diagonals are (3+1)x+(3−1)y=0 and (3−1)x−(3+1)y+83=0. Then a2 is equal to

Recall that the diagonals of a square are perpendicular bisectors of each other. Also, the distance from a vertex to the opposite diagonal is half the length of the diagonal.

Step 1: Identify Diagonals and their Properties✦ Active

First, let's find the slopes of the given lines. For L1:(3+1)x+(3−1)y=0, the slope is m1=−3+13−1=−(3+1)2(3−1)(3+1)=−3+1+233−1=−(2+3). For L2:(3−1)x−(3+1)y+83=0, the slope is m2=−3−1−(3+1)=3−13+1=(3−1)2(3+1)(3−1)=3+1−233−1=2−3. Since m1m2=−(2+3)(2−3)=−(4−3)=−1, the lines are perpendicular, confirming they are the diagonals of a square. The first equation L1 passes through the origin (0,0), so it represents the diagonal OB. The second equation L2 represents the diagonal AC.

Step 2: Calculate the Distance from Origin to Diagonal AC○ Expand

The distance from the origin O(0,0) to the diagonal AC (equation L2:(3−1)x−(3+1)y+83=0) is half the length of the diagonal. Using the distance formula from a point (x0,y0) to a line Ax+By+C=0:

D=|(3−1)(0)−(3+1)(0)+83|(3−1)2+(−(3+1))2 D=83(3−23+1)+(3+23+1)=834−23+4+23=838=8322=26

This distance D=26 is half the length of the diagonal AC.

💡 Teacher's Secret Hint

Remember that for a square with one vertex at the origin, the distance from the origin to the opposite diagonal is half the diagonal's length.

Step 3: Determine Side Length and a2○ Expand

The full length of the diagonal is d=2D=2(26)=46. For a square with side length a, the length of the diagonal is a2. Therefore, we can set up the equation:

a2=46 a=462=43

Finally, we need to find a2:

a2=(43)2=16×3=48
💡 Teacher's Secret Hint

Ensure to square both the numerical and radical parts of the side length a correctly.

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