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

The given points are $A(2, 4, 3)$, $B(4, a, 9)$, and $C(10, -1, 7)$.

The direction ratios of the line segment $AB$ are $(2 - 4, 4 - a, 3 - 9) = (-2, 4 - a, -6)$.

The direction ratios of the line segment $BC$ are $(10 - 4, -1 - a, 7 - 9) = (6, -1 - a, -2)$.

Since $\angle B = 90^\circ$, the line segments $AB$ and $BC$ are perpendicular to each other. Therefore, the dot product of their direction vectors is zero.

$(-2)(6) + (4 - a)(-1 - a) + (-6)(-2) = 0$

$-12 - (4 - a)(1 + a) + 12 = 0$

$(a - 4)(a + 1) = 0$

$a = 4$ or $a = -1$

Answer: $-1$ or $4$