Question

Find the coordinates of the point P where the line through A (3, -4 , -5 ) and B (2, -3 , 1) crosses the plane passing through three points L(2,2,1), M(3,0,1) and N(4, -1,0 ) . Also, find the ratio in which P diveides the line segment AB.

Answer 1

Equation of the plane passing through the points L(2, 2, 1),  M(3, 0, 1) and N(4, −1, 0) is

\[\left[ \vec{r} - \left( 2 \hat{i}  + 2 \hat{j} + \hat{k} \right) \right] . \left[ \left( \hat{i} - 2 \hat{j} \right) \times \left( \hat{i} - \hat{j} - \hat{k} \right) \right] = 0\]

\[ \Rightarrow \left[ \vec{r} - \left( 2 \hat{i} + 2 \hat{j} + \hat{k} \right) \right] . \left( 2 \hat{i} + \hat{j} + \hat{k} \right) = 0\]

\[ \Rightarrow \vec{r} . \left( 2 \hat{i} + \hat{j} + \hat{k} \right) = \left( 2 \hat{i} + 2 \hat{j} + \hat{k} \right) . \left( 2 \hat{i}  + \hat{j}  + \hat{k}  \right)\]

\[ \Rightarrow \vec{r} . \left( 2 \hat{i} + \hat{j} + \hat{k} \right) = 4 + 2 + 1 = 7 . . . . . \left( 1 \right)\]

The equation of line segment through A(3, −4, −5) and B(2, −3, 1) is

\[\frac{x - 3}{2 - 3} = \frac{y + 4}{- 3 + 4} = \frac{z + 5}{1 + 5}\]
\[i . e . \frac{x - 3}{- 1} = \frac{y + 4}{1} = \frac{z + 5}{6}\]

Any point on this line is of the form \[\left( - \lambda + 3, \lambda - 4, 6\lambda - 5 \right)\]

This point lies on the plane (1).

\[\therefore \left[ \left( - \lambda + 3 \right) \hat{i} + \left( \lambda - 4 \right) \hat{j} + \left( 6\lambda - 5 \right) \hat{k}  \right] . \left( 2 \hat{i}  + \hat{j}  + \hat{k}  \right) = 7\]
\[ \Rightarrow 2\left( - \lambda + 3 \right) + \left( \lambda - 4 \right) + \left( 6\lambda - 5 \right) = 7\]
\[ \Rightarrow 5\lambda = 10\]
\[ \Rightarrow \lambda = 2\]

Thus, the coordinates of the point P are (−2 + 3, 2 − 4, 6 × 2 − 5) i.e. (1, −2, 7).
Suppose P divides the line segment AB in the ratio μ : 1.

\[\therefore \left( 1, - 2, 7 \right) = \left( \frac{2\mu + 3}{\mu + 1}, \frac{- 3\mu - 4}{\mu + 1}, \frac{\mu - 5}{\mu + 1} \right)\]

\[ \Rightarrow \frac{2\mu + 3}{\mu + 1} = 1, \frac{- 3\mu - 4}{\mu + 1} = - 2, \frac{\mu - 5}{\mu + 1} = 7\]

\[ \Rightarrow 2\mu + 3 = \mu + 1, - 3\mu - 4 = - 2\mu - 2, \mu - 5 = 7\mu + 7\]

\[ \Rightarrow \mu = - 2\]

Thus, the point P divides the line segment AB externally in the ratio 2 : 1.