CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q252 Marksperpendicular_lines
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Question 25 (Three Dimensional Geometry)
Find p if (x−3)/1=(1−y)/1=(z+2)/p and (2−x)/3=(y+1)/5=(z+56)/(2p) are perpendicular. OR Find the vector equation of the line through the origin perpendicular to r=(2,−1,2)+λ(3,4,2) and r=μ(1,−1,1).
Alternative 1 2 Marks
Find p if (x−3)/1=(1−y)/1=(z+2)/p and (2−x)/3=(y+1)/5=(z+56)/(2p) are perpendicular.
Answer: p=±2.
- Read the direction ratios (1,−1,p) and (−3,5,2p).
- Set their dot product −3−5+2p² to zero.
- This gives p²=4 and p=±2.
Alternative 2 2 Marks
Find the vector equation of the line through the origin perpendicular to r=(2,−1,2)+λ(3,4,2) and r=μ(1,−1,1).
Answer: r=t(6,−1,−7).
- The line direction must be perpendicular to (3,4,2) and (1,−1,1).
- Their cross product is (6,−1,−7), which is nonzero.
- Use the origin as base point to get r=t(6,−1,−7).