Question

The ratio in which the plane $\overline{r}.(\overline{i}-2\overline{j}+3\overline{k})=17$ divides the line joining the points $-2\overline{i}+4\overline{j}+7\overline{k}$ and $3\overline{i}-5\overline{j}+8\overline{k}$ is

• A. $1:10$
• B. $3:10$
• C. $3:5$
• D. $1:5$

Answer 1

The correct option is B $3:10$

Let $\overrightarrow{r}$ be a point on a plane $\overrightarrow{r}.(\overrightarrow{i}-2\overrightarrow{j}+3\overrightarrow{k})=17$ which divides line joining the points $-2\overrightarrow{i}+4\overrightarrow{j}+7\overrightarrow{k}$ & $3\overrightarrow{i}-5\overrightarrow{j}+8\overrightarrow{k}$ in the ratio $n:m$
Therefore, $\overrightarrow{r}=\overrightarrow{r}=\frac{m(-2\overrightarrow{i}+4\overrightarrow{j}+7\overrightarrow{k})+n(3\overrightarrow{i}-5\overrightarrow{j}+8\overrightarrow{k})}{m+n}=\frac{(-2m+3n)i+(4m-5n)j+(7m+8n)k}{m+n}$
Substituting $\overrightarrow{r}$ in equation of plane, we get
$\left(\frac{(-2m+3n)\overrightarrow{i}+(4m-5n)\overrightarrow{j}+(7m+8n)\overrightarrow{k}}{m+n}\right).(\overrightarrow{i}-2\overrightarrow{j}+3\overrightarrow{k})=17$
$\Rightarrow -2m+3n-8m+10n+21m+24n=17m+17n$
$\Rightarrow -6m+20n=0$
$\Rightarrow n:m=3:10$
Ans: B