Question

If \[ \vec{OA}=\hat i-6\hat j+9\hat k,\qquad \vec{OB}=\hat i+3\hat j+5\hat k, \qquad \vec{OC}=2\hat i+\beta\hat j+7\hat k \] are the position vectors of three collinear points \(A,B,C\), then the ratio in which \(B\) divides \(AC\) is

• A. \(2:1\) externally
• B. \(1:2\) internally
• C. \(1:2\) externally
• D. \(2:1\) internally

Hint:

For collinear points, if \[ \overrightarrow{AB}=k\overrightarrow{BC} \] with \(k<0\), the point \(B\) divides the line externally. The absolute value of \(k\) gives the ratio.

Answer 1

The Correct Option is A

Concept: When three points are collinear, one point can be expressed as a linear combination of the other two. For external division, the section formula in vector form is \[ \vec{OB} = \frac{m\vec{OC}-n\vec{OA}}{m-n}. \] To determine the ratio, we compare the coordinates of the vectors and identify the values of \(m\) and \(n\).

Step 1: Write the coordinates of the given points. \[ A=(1,-6,9), \] \[ B=(1,3,5), \] \[ C=(2,\beta,7). \] Since \(A,B,C\) are collinear, \[ \overrightarrow{AB} = B-A = (0,9,-4), \] and \[ \overrightarrow{BC} = C-B = (1,\beta-3,2). \] For collinearity, \[ \overrightarrow{AB} = \lambda\overrightarrow{BC}. \]

Step 2: Determine the parameter. Comparing the \(z\)-components, \[ -4=2\lambda \] gives \[ \lambda=-2. \] Now compare the \(y\)-components: \[ 9=-2(\beta-3). \] \[ 9=-2\beta+6. \] \[ -2\beta=3. \] \[ \beta=-\frac32. \]

Step 3: Interpret the value of \(\lambda\). Since \[ \overrightarrow{AB} = -2\overrightarrow{BC}, \] the negative sign indicates that \(B\) lies outside the segment \(AC\), i.e. the division is external. The magnitude relation gives \[ AB:BC=2:1. \] Hence \(B\) divides \(AC\) externally in the ratio \[ 2:1. \] Conclusion: \[ \boxed{2:1 \text{ externally}} \]