The straight lines ${\mathrm{\ell }}_1\mathrm{\&}{\mathrm{\ell }}_2$ pass through the origin and trisect the line segment of the line L: 9x + 5y = 45 between the axes. If  ${\mathrm{m}}_{1 }\mathrm{\&} {\mathrm{m}}_2$ are the slopes of the lines ${\mathrm{\ell }}_1\mathrm{\&}{\mathrm{\ell }}_2,$ then the point of intersection of the line $\mathrm{y}=\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{x}$ with L.

Given ${\mathrm{\ell }}_1,{\mathrm{\ell }}_2$ trisects the given line between coordinate axis. The points P, M are points of trisections.

From section formula (m : n) 1 : 2 or 2 : 1

The points be $\left(\frac{5}{3},6\right)\mathrm{\&}\left(\frac{10}{3},3\right)$      $\therefore$The lines are

$$\begin{gathered} \mathrm{y}=\frac{18}{5}\mathrm{x} \\ \mathrm{y}=\frac{9}{10}\mathrm{x} \\ \Rightarrow {\mathrm{m}}_1=\frac{18}{5},{\mathrm{m}}_2=\frac{9}{10} \end{gathered}$$

$\therefore \text{ The line }\mathrm{y}=\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{x}\Rightarrow \mathrm{y}=\frac{9}{2}\mathrm{x}$

$\because \text{ The line intersects with the given lineL: 9x + 5y = 45at }\left(\frac{10}{7},\frac{45}{7}\right)$

From the options y – x = 5 passes through above point.