CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Q303 Markspoint_line_distance
CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Question 30 (Three Dimensional Geometry)
Find the points on (x−2)/3=(1−y)/2=(z−3)/2 at distance √2 from (1,2,3). OR Find the shortest distance between r=(4,−1,0)+λ(1,2,−3) and r=(1,−1,2)+μ(2,4,−5).
Alternative 1 3 Marks
Find the points on (x−2)/3=(1−y)/2=(z−3)/2 at distance √2 from (1,2,3).
Answer: (2,1,3) and (4/17,37/17,31/17).
- Write a point as (2+3t,1−2t,3+2t).
- Its squared distance from (1,2,3) is (1+3t)²+(−1−2t)²+4t². Set this to 2 to obtain 17t²+10t=0.
- Substitute t=0 and t=−10/17 to get both points.
Alternative 2 3 Marks
Find the shortest distance between r=(4,−1,0)+λ(1,2,−3) and r=(1,−1,2)+μ(2,4,−5).
Answer: 6/√5.
- The cross product of the two line directions is (2,−1,0).
- The displacement between base points is (−3,0,2). Its dot product with the normal is −6.
- Divide the absolute dot product by the normal length √5.