Question

The ratio in which the line segment joining the points whose position vectors are $2\hat{i}-4\hat{j}-7\hat{k}$ and $-3\hat{i}+5\hat{j}-8\hat{k}$ is divided by the plane whose equation is $\hat{r}.(\hat{i}-2\hat{j}+3\hat{k})=13$ is

• A. $13:12$ internally
• B. $12:25$ externally
• C. $13:25$ internally
• D. $37:25$ internally

Answer 1

The correct option is B $12:25$ externally

Let $P$ be the point and it divides the line segment in the ratio $\lambda :1.$ Then,
$\overrightarrow{OP}=\overrightarrow{r}=\frac{-3\lambda +2}{\lambda +1}\hat{i}+\frac{5\lambda -4}{\lambda +1}\hat{j}+\frac{-8\lambda -7}{\lambda +1}\hat{k}$

It satisfies $\overrightarrow{r}.(\hat{i}-2\hat{j}+3\hat{k})=13.$
$\therefore \frac{-3\lambda +2}{\lambda +1}-2\frac{5\lambda -4}{\lambda +1}+3\frac{-8\lambda -7}{\lambda +1}=13$
$\Rightarrow -3\lambda +2-2(5\lambda -4)+3(-8\lambda -7)=13(\lambda +1)$
$\Rightarrow -37\lambda -11=13\lambda +13$
$\Rightarrow \lambda =-\frac{12}{25}$