The position vectors of the points A and B with respect to O are 2i^+2j^+k^ and 2i^+4j^+4k^. The length of the internal bisector of ∠BOA of ∆AOB is (take proportionality constant is 2)

AP EAMCET · Maths · Three Dimensional Geometry

The position vectors of the points $A$ and $B$ with respect to $O$ are $2 \hat{i} + 2 \hat{j} + \hat{k}$ and $2 \hat{i} + 4 \hat{j} + 4 \hat{k}$ . The length of the internal bisector of $∠ B O A$ of $∆ A O B$ is (take proportionality constant is $2$ )

A $\frac{\sqrt{136}}{9}$
B $\frac{\sqrt{136}}{3}$
C $\frac{20}{3}$
D $\frac{25}{3}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{\sqrt{136}}{3}$

Step-by-step Solution

Detailed explanation

Here OA=22+22+12=3, OB=22+42+42=6 So, Internal bisector OD divides AB in the ratio =1:2, using angle bisector theorem Using section formula, we get Position vectors of D are (2, 83,2)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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