Question

Let O be the origin and the position vector 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

• A. \(\frac{2}{3}\sqrt31\)
• B. \(\frac{2}{3}\sqrt34\)
• C. \(\frac{3}{4}\sqrt34\)
• D. \(\frac{3}{2}\sqrt31\)

Answer 1

The Correct Option is B

The given problem involves determining the point where the internal angle bisector of \(\angle AOB\) meets line AB. Let's solve this geometrically using vectors. 

Given:

• Position vector of A: \(\mathbf{a} = 2\hat{i} + 2\hat{j} + \hat{k}\)
• Position vector of B: \(\mathbf{b} = 2\hat{i} + 4\hat{j} + 4\hat{k}\)
• Origin O: \(\mathbf{O} = 0\hat{i} + 0\hat{j} + 0\hat{k}\)

To find the length OC, where C is the point of intersection of the angle bisector of \(\angle AOB\) with the line AB, follow these steps:

1. Calculate the vector \(\mathbf{a} - \mathbf{O}\) and \(\mathbf{b} - \mathbf{O}\):
   • \(\mathbf{a} = 2\hat{i} + 2\hat{j} + \hat{k}\)
   • \(\mathbf{b} = 2\hat{i} + 4\hat{j} + 4\hat{k}\)
2. Calculate the magnitudes of these vectors:
   • \(|\mathbf{a}| = \sqrt{(2)^2 + (2)^2 + (1)^2} = \sqrt{9} = 3\)
   • \(|\mathbf{b}| = \sqrt{(2)^2 + (4)^2 + (4)^2} = \sqrt{36} = 6\)
3. Use the internal bisector theorem, which states:

The point C dividing the segment AB in the ratio \(|\mathbf{b}|:|\mathbf{a}|\) will be on the angle bisector of \(\angle AOB\).

• The ratio is \(6:3\) or \(2:1\).

1. Determine the position vector of C using the section formula:
   • \(\mathbf{r}_c = \frac{2(2\hat{i} + 2\hat{j} + \hat{k}) + 1(2\hat{i} + 4\hat{j} + 4\hat{k})}{2 + 1}\)
   • \(\mathbf{r}_c = \frac{(4\hat{i} + 4\hat{j} + 2\hat{k}) + (2\hat{i} + 4\hat{j} + 4\hat{k})}{3}\)
   • \(\mathbf{r}_c = \frac{6\hat{i} + 8\hat{j} + 6\hat{k}}{3} = 2\hat{i} + \frac{8}{3}\hat{j} + 2\hat{k}\)
2. Calculate the length of OC:
   • \(|\mathbf{OC}| = \sqrt{(2)^2 + \left(\frac{8}{3}\right)^2 + (2)^2}\)
   • \(|\mathbf{OC}| = \sqrt{4 + \frac{64}{9} + 4}\)
   • \(|\mathbf{OC}| = \sqrt{\frac{72 + 64}{9}}\)
   • \(|\mathbf{OC}| = \sqrt{\frac{136}{9}} = \frac{\sqrt{136}}{3}\)
   • \(|\mathbf{OC}| = \frac{2\sqrt{34}}{3}\)

Therefore, the length of OC is \(\frac{2}{3}\sqrt{34}\). This matches with the given correct option: \(\frac{2}{3}\sqrt{34}\).

Answer 2

Step 1: Find the Coordinates of Points A and B

A = (2, 2, 1) and B = (2, 4, 4)

Step 2: Use the Internal Division Formula

The internal bisector of ∠AOB divides AB in the ratio OA : OB = 1 : 2. Using the section formula, the coordinates of C are:

\[ C = \frac{1 \cdot B + 2 \cdot A}{1 + 2} = \frac{1 \cdot (2, 4, 4) + 2 \cdot (2, 2, 1)}{3} = \left(2, \frac{8}{3}, 2\right) \]

Step 3: Calculate the Length of OC

The vector OC has coordinates (2, \(\frac{8}{3}\), 2). Using the distance formula:

\[ |OC| = \sqrt{2^2 + \left(\frac{8}{3}\right)^2 + 2^2} = \sqrt{4 + \frac{64}{9} + 4} = \sqrt{\frac{136}{9}} = \frac{2\sqrt{34}}{3} \]

So, the correct answer is: \(\frac{2}{3}\sqrt{34}\)