Question

The coordinates of the point which divides the line joining the points $ (2, 3, 4) $ and $ (3, -4, 7) $ in the ratio 2 : 4 externally is:

• A. \( (10, 1, 1) \)
• B. \( (1, 10, 1) \)
• C. \( (10, -10, 10) \)
• D. \( (1, 1, 10) \)

Hint:

For external division, use the section formula with a negative sign for the denominator and solve for each coordinate separately.

Answer 1

The Correct Option is A

Let the coordinates of the point which divides the line joining \( A(2, 3, 4) \) and \( B(3, -4, 7) \) in the ratio 2 : 4 externally be \( P(x, y, z) \).
We use the section formula for external division: \[ x = \frac{m x_2 - n x_1}{m - n}, \quad y = \frac{m y_2 - n y_1}{m - n}, \quad z = \frac{m z_2 - n z_1}{m - n} \] where \( m = 2 \) and \( n = 4 \), and the coordinates of \( A \) and \( B \) are \( (2, 3, 4) \) and \( (3, -4, 7) \), respectively. Step 1: Calculate the x-coordinate. \[ x = \frac{2(3) - 4(2)}{2 - 4} = \frac{6 - 8}{-2} = \frac{-2}{-2} = 1 \] Step 2: Calculate the y-coordinate. \[ y = \frac{2(-4) - 4(3)}{2 - 4} = \frac{-8 - 12}{-2} = \frac{-20}{-2} = 10 \] Step 3: Calculate the z-coordinate. \[ z = \frac{2(7) - 4(4)}{2 - 4} = \frac{14 - 16}{-2} = \frac{-2}{-2} = 1 \] Thus, the coordinates of the point are \( \boxed{(10, 1, 1)} \).