3D Geometry
Angle Bisector Meets AB — Length OC
nta_pyq_2024_jan
Grade 12
Question:
Let O be the origin and the position vectors of $A$ and $B$ be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$, then the length of $OC$ is
$\dfrac{2}{3}\sqrt{31}$
$\dfrac{2}{3}\sqrt{34}$
$\dfrac{3}{4}\sqrt{34}$
$\dfrac{3}{2}\sqrt{31}$
Step-by-Step Solution
Key Concept: $OA=3$, $OB=6$. Internal bisector divides $AB$ in ratio $OA:OB=3:6=1:2$. $C=\frac{2A+1B}{3}=\frac{(4+2,4+4,2+4)}{3}=(2,8/3,2)$.
$OC=\frac{2\sqrt{34}}{3}$.
Correct Answer: 2