CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q325 Marksline_intersection
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Question 32 (Three Dimensional Geometry)
Write (x+3)/(−3)=(y−1)/1=(z−5)/5 and (x+1)/(−1)=(2−y)/(−2)=(z−5)/5 in vector form and check whether the lines intersect. OR Opposite sides of a square lie along r=(1,2,−4)+λ(2,3,6) and r=(3,3,−5)+μ(2,3,6). The other pair has direction ratios (−3,6,p). Find the area and p.
Alternative 1 5 Marks
Write (x+3)/(−3)=(y−1)/1=(z−5)/5 and (x+1)/(−1)=(2−y)/(−2)=(z−5)/5 in vector form and check whether the lines intersect.
Answer: r=(1+s)(−3,1,5) and r=(1+t)(−1,2,5). They meet at the origin.
- The first line has base point and direction both equal to (−3,1,5).
- The second has base point and direction both equal to (−1,2,5).
- Both contain the origin at s=t=−1. Their directions are nonparallel, so they intersect uniquely there.
Alternative 2 5 Marks
Opposite sides of a square lie along r=(1,2,−4)+λ(2,3,6) and r=(3,3,−5)+μ(2,3,6). The other pair has direction ratios (−3,6,p). Find the area and p.
Answer: Area 293/49 square units and p=−2.
- The displacement between base points is (2,1,−1). Its cross product with (2,3,6) is (9,−14,4), of norm √293.
- The parallel-line distance is √293/7, the side length. Its square is 293/49.
- Adjacent directions are perpendicular, so −6+18+6p=0 and p=−2.