Let O be the origin and the position vectors of A and B be and respectively. If the internal bisector of meets the line AB at C, then the length of OC is [2024]

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Solution

Q.
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 $∠ A O B$ meets the line AB at C, then the length of OC is [2024]

1 $\frac{2}{3} \sqrt{31}$

2 $\frac{3}{2} \sqrt{34}$

3 $\frac{2}{3} \sqrt{34}$

4 $\frac{3}{2} \sqrt{31}$

Ans.
(3)

$| \vec{O A} |$ $= \sqrt{4 + 4 + 1} = 3$

$| \vec{O B} |$ $= \sqrt{4 + 16 + 16} = 6$

$\frac{O A}{O B} = \frac{A C}{C B} = \frac{3}{6} = \frac{1}{2}$

$C \equiv (\frac{4 + 2}{3}, \frac{4 + 4}{3}, \frac{2 + 4}{3})$

$∴ C \equiv (2, \frac{8}{3}, 2)$

$| \vec{O C} |$ $= \sqrt{4 + \frac{64}{9} + 4} = \sqrt{\frac{136}{3}} = \frac{2 \sqrt{34}}{3}$ .

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