CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q355 Marksthree_dimensional_geometry

CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 35 (Three Dimensional Geometry)

Check whether the lines given by (x−1)/2 = (y−2)/3 = (z−3)/4 and (x−4)/5 = (y−1)/2 = z, are parallel or not. If parallel, find distance between them, otherwise if the lines intersect, find their point of intersection.

✅ Step-by-Step Solution

Final Answer: Lines intersect at (−1, −1, −1).

Direction ratios of L1: (2,3,4), L2: (5,2,1). Since not proportional, lines are not parallel.
Parametric: L1: (1+2t, 2+3t, 3+4t); L2: (4+5s, 1+2s, s). Equating: 1+2t = 4+5s, 2+3t = 1+2s, 3+4t = s. Solve: from third s=3+4t. Substitute in first: 1+2t = 4+5(3+4t) = 19+20t ⇒ −18=18t ⇒ t=−1, s=−1. Check second: 2+3(-1)= −1, 1+2(-1)=−1, consistent. Point: (1+2(−1), 2+3(−1), 3+4(−1)) = (−1, −1, −1).
Lines intersect at (−1, −1, −1).

📋 CBSE Stepwise Marking Scheme

[1 mark] Direction ratios of L1: (2,3,4), L2: (5,2,1). Since not proportional, lines are not parallel.
[3 marks] Parametric: L1: (1+2t, 2+3t, 3+4t); L2: (4+5s, 1+2s, s). Equating: 1+2t = 4+5s, 2+3t = 1+2s, 3+4t = s. Solve: from third s=3+4t. Substitute in first: 1+2t = 4+5(3+4t) = 19+20t ⇒ −18=18t ⇒ t=−1, s=−1. Check second: 2+3(-1)= −1, 1+2(-1)=−1, consistent. Point: (1+2(−1), 2+3(−1), 3+4(−1)) = (−1, −1, −1).
[1 mark] Lines intersect at (−1, −1, −1).