Class 12 Three Dimensional Geometry · Very Long Answer5 Marksline_intersections
Class 12 Maths · Three Dimensional Geometry · Very Long Answer #92
Find equations of diagonals and their intersection for parallelogram P(4,2,−6),Q(5,−3,1),R(12,4,5),S(11,9,−2).
Parallelogram diagonals and common midpoint
✅ Step-by-Step Solution
Final Answer: PR: r=(4,2,−6)+t(8,2,11); QS: r=(5,−3,1)+u(6,12,−3); intersection (8,3,−1/2).
- Compute R−P=(8,2,11), S−Q=(6,12,−3).
- PR has vector equation r=P+t(8,2,11), Cartesian (x−4)/8=(y−2)/2=(z+6)/11.
- QS has vector equation r=Q+u(6,12,−3), Cartesian (x−5)/6=(y+3)/12=(z−1)/(−3).
- The common midpoint is (P+R)/2=(Q+S)/2=(8,3,−1/2).
- Both diagonal parameters are 1/2, so the intersection is (8,3,−1/2).
📋 CBSE Stepwise Marking Scheme
- [1 mark] Compute R−P=(8,2,11), S−Q=(6,12,−3).
- [1 mark] PR has vector equation r=P+t(8,2,11), Cartesian (x−4)/8=(y−2)/2=(z+6)/11.
- [1 mark] QS has vector equation r=Q+u(6,12,−3), Cartesian (x−5)/6=(y+3)/12=(z−1)/(−3).
- [1 mark] The common midpoint is (P+R)/2=(Q+S)/2=(8,3,−1/2).
- [1 mark] Both diagonal parameters are 1/2, so the intersection is (8,3,−1/2).
