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

$$\begin{gathered} \left[\overrightarrow{r}-\left(2\hat{i}+2\hat{j}+\hat{k}\right)\right].\left[\left(\hat{i}-2\hat{j}\right)\times \left(\hat{i}-\hat{j}-\hat{k}\right)\right]=0 \\ \Rightarrow \left[\overrightarrow{r}-\left(2\hat{i}+2\hat{j}+\hat{k}\right)\right].\left(2\hat{i}+\hat{j}+\hat{k}\right)=0 \\ \Rightarrow \overrightarrow{r}.\left(2\hat{i}+\hat{j}+\hat{k}\right)=\left(2\hat{i}+2\hat{j}+\hat{k}\right).\left(2\hat{i}+\hat{j}+\hat{k}\right) \\ \Rightarrow \overrightarrow{r}.\left(2\hat{i}+\hat{j}+\hat{k}\right)=4+2+1=7.....\left(1\right) \end{gathered}$$
The equation of line segment through A(3, −4, −5) and B(2, −3, 1) is

$$\begin{gathered} \frac{x-3}{2-3}=\frac{y+4}{-3+4}=\frac{z+5}{1+5} \\ \mathrm{i}.\mathrm{e}.\frac{{\displaystyle x-3}}{{\displaystyle -1}}=\frac{{\displaystyle y+4}}{{\displaystyle 1}}=\frac{{\displaystyle z+5}}{{\displaystyle 6}} \end{gathered}$$
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).

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

$$\begin{gathered} \therefore \left(1,-2,7\right)=\left(\frac{2\mu +3}{\mu +1},\frac{-3\mu -4}{\mu +1},\frac{\mu -5}{\mu +1}\right) \\ \Rightarrow \frac{2\mu +3}{\mu +1}=1,\frac{-3\mu -4}{\mu +1}=-2,\frac{\mu -5}{\mu +1}=7 \\ \Rightarrow 2\mu +3=\mu +1,-3\mu -4=-2\mu -2,\mu -5=7\mu +7 \\ \Rightarrow \mu =-2 \end{gathered}$$
Thus, the point P divides the line segment AB externally in the ratio 2 : 1.