Question
ABC is a triangle right-angled at B. If A(k, 1, -1), B(2k, 0, 2) and C(2+2k, k, 1) are the vertices of the triangle, then what is the value of k?
NDA
Mathematics
Vector Algebra
2025
NDA 2025 (Phase 1)
NDA Mathematics Question (2025) — Solution
Options
- A. -3
- B. -1
- C. 1
- D. 3
Answer
D. 3
Step-by-step solution
The direction ratios of the line segment AB are (k - 2k, 1 - 0, -1 - 2) = (-k, 1, -3). The direction ratios of the line segment BC are (2+2k - 2k, k - 0, 1 - 2) = (2, k, -1). Since the triangle is right-angled at B, the line segments AB and BC are perpendicular to each other. Therefore, the dot product of their direction ratios is zero. (-k)(2) + (1)(k) + (-3)(-1) = 0 -2k + k + 3 = 0 k = 3 Answer: 3
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