Question

The three points $A(2, 4, 3)$, $B(4, a, 9)$ and $C(10, -1, 7)$ form a right-angled triangle with $\angle B = 90^\circ$, then the value of 'a' is

• A. 1 or 4
• B. -2 or 4
• C. 1 or -4
• D. -2 or -4

Hint:

For right-angled triangle problems with coordinates, setting the dot product of the perpendicular side vectors to zero is always the fastest and most direct method. Avoid using the full distance formula unless necessary.

Answer 1

The Correct Option is B

Step 1: Use right angle condition
Since $\angle B = 90^\circ$, vectors $\vec{BA}$ and $\vec{BC}$ are perpendicular: \[ \vec{BA} \cdot \vec{BC} = 0 \]
Step 2: Find vectors
\[ \vec{BA} = A - B = (2-4,\; 4-a,\; 3-9) = (-2,\; 4-a,\; -6) \] \[ \vec{BC} = C - B = (10-4,\; -2-a,\; 7-9) = (6,\; -2-a,\; -2) \]
Step 3: Apply dot product
\[ (-2)(6) + (4-a)(-2-a) + (-6)(-2) = 0 \] \[ -12 + (4-a)(-2-a) + 12 = 0 \]
Step 4: Simplify
\[ (4-a)(-2-a) = 0 \] \[ -(a-4)(a+2) = 0 \] \[ (a-4)(a+2) = 0 \]
Step 5: Solve
\[ a = 4 \text{or} a = -2 \] Final Answer:
\[ \boxed{a = -2 \text{ or } 4} \]