Question

Find the ratio in which the line segment joining the points (2, –1, 3) and (–1, 2, 1) is divided by the plane x + y + z = 5. 

Answer 1

Suppose the plane x + y + z = 5 divides the line joining the points A (2, −1, 3) and B (\[-\]1, 2, 1) at a point C in the ratio\[\lambda: 1\]

Then, coordinates of C are as follows:

\[\left( \frac{- \lambda + 2}{\lambda + 1}, \frac{2\lambda - 1}{\lambda + 1}, \frac{\lambda + 3}{\lambda + 1} \right)\]

Now, the point C lies on the plane x + y + z = 5.
Therefore, the coordinates of C must satisfy the equation of the plane.

\[\frac{- \lambda + 2}{\lambda + 1} + \frac{2\lambda - 1}{\lambda + 1} + \frac{\lambda + 3}{\lambda + 1} = 5\]

\[ \Rightarrow - \lambda + 2 + 2\lambda - 1 + \lambda + 3 = 5\lambda + 5\]

\[ \Rightarrow 2\lambda + 4 = 5\lambda + 5\]

\[ \Rightarrow - 3\lambda = 1\]

\[ \therefore \lambda = \frac{- 1}{3}\]

So, the required ratio is 1:3 (externally).