NDA 1 2025 Maths, question 63

Topic: Analytical geometry

Q63

$ABC$ 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 $k$ ?

(a) $- 3$
(b) $- 1$
(c) $1$
(d) $3$ Correct answer

Solution.

Answer: (d) $3$

Perpendicular sides at $B$ require $\overrightarrow{BA} \cdot \overrightarrow{BC} = 0$ .
Here $\overrightarrow{BA} = ( - k,1, - 3)$ and $\overrightarrow{BC} = (2,k, - 1)$ .
The dot product is $- 2k + k + 3 = 3 - k$ .
Hence $k = 3$ .