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:
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:
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In three-dimensional geometry, the section formula is applied coordinate-wise. For external division, the ratio may contain a negative sign.
Updated On: Jun 26, 2026
- \((6,7,8)\)
- \((5,6,7)\)
- \((-4,-5,-6)\)
- \((-5,-6,-7)\)
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The Correct Option is B
Solution and Explanation
Step 1: Use section formula to find \(\lambda\).
The point \((3,4,5)\) divides the line segment joining \[ (1,2,3) \] and \[ (4,5,6) \] in the ratio \(\lambda:1\). Using section formula, \[ (3,4,5) = \left( \frac{4\lambda+1}{\lambda+1}, \frac{5\lambda+2}{\lambda+1}, \frac{6\lambda+3}{\lambda+1} \right). \] Using the first coordinate, \[ 3=\frac{4\lambda+1}{\lambda+1}. \] So, \[ 3\lambda+3=4\lambda+1. \] Hence, \[ \lambda=2. \]
Step 2: Use the second given ratio.
Now we need the point which divides the line segment joining \[ (3,4,5) \] and \[ (1,2,3) \] in the ratio \[ -1:\lambda. \] Since \[ \lambda=2, \] the ratio is \[ -1:2. \]
Step 3: Apply section formula.
Let the required point be \(P\). Then \[ P= \left( \frac{-1(1)+2(3)}{-1+2}, \frac{-1(2)+2(4)}{-1+2}, \frac{-1(3)+2(5)}{-1+2} \right). \] Therefore, \[ P= \left( \frac{-1+6}{1}, \frac{-2+8}{1}, \frac{-3+10}{1} \right). \] Thus, \[ P=(5,6,7). \]
Step 4: Final conclusion.
Hence, the required point is \[ \boxed{(5,6,7)}. \]
The point \((3,4,5)\) divides the line segment joining \[ (1,2,3) \] and \[ (4,5,6) \] in the ratio \(\lambda:1\). Using section formula, \[ (3,4,5) = \left( \frac{4\lambda+1}{\lambda+1}, \frac{5\lambda+2}{\lambda+1}, \frac{6\lambda+3}{\lambda+1} \right). \] Using the first coordinate, \[ 3=\frac{4\lambda+1}{\lambda+1}. \] So, \[ 3\lambda+3=4\lambda+1. \] Hence, \[ \lambda=2. \]
Step 2: Use the second given ratio.
Now we need the point which divides the line segment joining \[ (3,4,5) \] and \[ (1,2,3) \] in the ratio \[ -1:\lambda. \] Since \[ \lambda=2, \] the ratio is \[ -1:2. \]
Step 3: Apply section formula.
Let the required point be \(P\). Then \[ P= \left( \frac{-1(1)+2(3)}{-1+2}, \frac{-1(2)+2(4)}{-1+2}, \frac{-1(3)+2(5)}{-1+2} \right). \] Therefore, \[ P= \left( \frac{-1+6}{1}, \frac{-2+8}{1}, \frac{-3+10}{1} \right). \] Thus, \[ P=(5,6,7). \]
Step 4: Final conclusion.
Hence, the required point is \[ \boxed{(5,6,7)}. \]
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