Question

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

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

Hint:

In three-dimensional geometry, the section formula is applied coordinate-wise. For external division, the ratio may contain a negative sign.

Answer 1

The Correct Option is B

Step 1: Find lambda using the x-coordinate.
Point \((3,4,5)\) divides \((1,2,3)\) and \((4,5,6)\) in ratio \(\lambda:1\). x-coord: \(\frac{4\lambda+1}{\lambda+1}=3\Rightarrow4\lambda+1=3\lambda+3\Rightarrow\lambda=2\).

Step 2: Divide (3,4,5) to (1,2,3) in ratio -1:2 (external).
\[x=\frac{(-1)(1)+(2)(3)}{-1+2}=\frac{5}{1}=5,\quad y=\frac{-2+8}{1}=6,\quad z=\frac{-3+10}{1}=7\] \[\boxed{(5,6,7)}\]