JEE Main 2024 — 29 January, Shift 1 — Question 12

Compute lengths OA and OB: $|OA| = \sqrt{2^2+2^2+1^2}=3$, $|OB| = \sqrt{2^2+4^2+4^2}=6$. By angle bisector theorem, $AC:CB = OA:OB = 1:2$. Using section formula for internal division, position vector of C is $\frac{2\, \vec{OA} + 1\, \vec{OB}}{1+2} = \frac{2\vec{a} + \vec{b}}{3}$. $2\vec{a} = (4,4,2)$, $\vec{b} = (2,4,4)$, so sum = $(6,8,6)$, thus $\vec{c} = (2,8/3,2)$. Length OC = $|\vec{c}| = \sqrt{2^2 + (8/3)^2 + 2^2} = \sqrt{4 + 64/9 + 4} = \sqrt{8 + 64/9}$. Simplify: $8 = 72/9$, so sum = $136/9$. Hence OC = $\sqrt{136/9} = \frac{\sqrt{136}}{3} = \frac{2\sqrt{34}}{3}$. Therefore, the length of OC is $\frac{2\sqrt{34}}{3}$.