Question

If the point \( (3,4,5) \) divides the line segment joining the points \( (1,2,3) \) and \( (4,5,6) \) in the ratio \( 2:1 \), then the point which divides the line segment joining the points \( (3,4,5) \) and \( (1,2,3) \) in the ratio \( -1:2 \) is:

• A. \( (6, 7, 8) \)
• B. \( (5, 6, 7) \)
• C. \( (-4, -5, -6) \)
• D. \( (-5, -6, -7) \)

Hint:

3D Section Formula}
Use: \( \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}, \frac{m z_2 + n z_1}{m + n} \right) \)
For negative ratio (external division), signs in numerator change
Always simplify before evaluating

Answer 1

The Correct Option is B

Let points be \( A = (3,4,5) \), \( B = (1,2,3) \). Ratio = \( -1:2 \) Using section formula for internal division: \[ P = \left( \frac{-1 \cdot 1 + 2 \cdot 3}{-1 + 2},\ \frac{-1 \cdot 2 + 2 \cdot 4}{-1 + 2},\ \frac{-1 \cdot 3 + 2 \cdot 5}{-1 + 2} \right) = \left( \frac{-1 + 6}{1}, \frac{-2 + 8}{1}, \frac{-3 + 10}{1} \right) = (5, 6, 7) \]