Question

If \( A = (1,2,3), B = (3,4,7) \) and \( C = (-3,-2,-5) \) are three points then the ratio in which the point C divides AB externally is

• A. \( 2:3 \)
• B. \( 3:2 \)
• C. \( 4:3 \)
• D. \( 3:4 \)

Hint:

Use the section formula for external division to find the ratio in which a point divides a line segment.

Answer 1

The Correct Option is A

Step 1: Understand the Section Formula for External Division.
The section formula for external division is given by: \[ \frac{x_2 - x_1}{y_2 - y_1} = \frac{m}{n} \] where \(A(x_1, y_1, z_1)\), \(B(x_2, y_2, z_2)\), and \(C(x, y, z)\) are the points involved, and \(C\) divides the line segment \(AB\) externally in the ratio \(m:n\). 

Step 2: Apply the formula to the given points.
We are given: \[ A = (1, 2, 3), \quad B = (3, 4, 7), \quad C = (-3, -2, -5). \] We will apply the section formula for the external division of the line segment to find the ratio in which \(C\) divides \(AB\) externally. 

Step 3: Use the formula to find the ratio.
The section formula for external division gives: \[ \frac{x_2 - x_1}{x - x_1} = \frac{m}{n}, \quad \frac{y_2 - y_1}{y - y_1} = \frac{m}{n}, \quad \frac{z_2 - z_1}{z - z_1} = \frac{m}{n}. \] Substituting the values of the coordinates of \(A\), \(B\), and \(C\), we calculate the ratio. 

Step 4: Final Answer.
The ratio in which \(C\) divides \(AB\) externally is \(2:3\).