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

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

It satisfies $\overrightarrow{\mathrm{r}}\cdot (\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+3\widehat{\mathrm{k}})=13\text{. So, }$

$\begin{array}{l}\frac{-3\mathrm{\lambda }+2}{\mathrm{\lambda }+1}-2\frac{5\mathrm{\lambda }-4}{\mathrm{\lambda }+1}+3\frac{-8\mathrm{\lambda }-7}{\mathrm{\lambda }+1}=13\\ \mathrm{or} -3\mathrm{\lambda }+2-2(5\mathrm{\lambda }-4)+3(-8\mathrm{\lambda }-7)=13(\mathrm{\lambda }+1)\\ \mathrm{or} -37\mathrm{\lambda }-11=13\mathrm{\lambda }+13\text{ or }50\mathrm{\lambda }=-24\text{ or }\mathrm{\lambda }=-\frac{12}{25}\end{array}$