The straight lines $$l_1$$ and $$l_2$$ pass through the origin and trisect the line segment of the line $$L: 9x + 5y = 45$$ between the axes. If $$m_1$$ and $$m_2$$ are the slopes of the lines $$l_1$$ and $$l_2$$, then the point of intersection of the line $$y = (m_1 + m_2)x$$ with L lies on

We need to find the point of intersection of a specific line with $$L: 9x + 5y = 45$$, and determine which of the given lines it lies on.

To begin,

The line $$L: 9x + 5y = 45$$ intersects the axes at:

- x-intercept: Set $$y = 0$$: $$9x = 45$$, so $$x = 5$$. Point $$A = (5, 0)$$.

- y-intercept: Set $$x = 0$$: $$5y = 45$$, so $$y = 9$$. Point $$B = (0, 9)$$.

Next,

The two points that trisect the segment AB divide it into three equal parts. Using the section formula, if a point divides the segment from $$A(x_1, y_1)$$ to $$B(x_2, y_2)$$ in the ratio $$m:n$$, then the point is $$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$$.

Point $$P_1$$ divides AB in ratio 1:2 (closer to A):

$$ P_1 = \left(\frac{1(0) + 2(5)}{3}, \frac{1(9) + 2(0)}{3}\right) = \left(\frac{10}{3}, 3\right) $$

Point $$P_2$$ divides AB in ratio 2:1 (closer to B):

$$ P_2 = \left(\frac{2(0) + 1(5)}{3}, \frac{2(9) + 1(0)}{3}\right) = \left(\frac{5}{3}, 6\right) $$

From here,

Line $$l_1$$ passes through the origin $$(0,0)$$ and $$P_1 = (10/3, 3)$$:

$$ m_1 = \frac{3 - 0}{10/3 - 0} = \frac{3}{10/3} = \frac{9}{10} $$

Line $$l_2$$ passes through the origin $$(0,0)$$ and $$P_2 = (5/3, 6)$$:

$$ m_2 = \frac{6 - 0}{5/3 - 0} = \frac{6}{5/3} = \frac{18}{5} $$

Continuing,

$$ m_1 + m_2 = \frac{9}{10} + \frac{18}{5} = \frac{9}{10} + \frac{36}{10} = \frac{45}{10} = \frac{9}{2} $$

Now,

The line $$y = \frac{9}{2}x$$ must be intersected with $$L: 9x + 5y = 45$$.

Substituting $$y = \frac{9}{2}x$$ into the equation of L:

$$ 9x + 5 \cdot \frac{9x}{2} = 45 $$

$$ 9x + \frac{45x}{2} = 45 $$

Multiplying through by 2:

$$ 18x + 45x = 90 $$

$$ 63x = 90 $$

$$ x = \frac{90}{63} = \frac{10}{7} $$

And $$y = \frac{9}{2} \cdot \frac{10}{7} = \frac{90}{14} = \frac{45}{7}$$

The point of intersection is $$\left(\frac{10}{7}, \frac{45}{7}\right)$$.

Moving forward,

Testing each option:

Option 3: $$y - x = 5$$

$$ \frac{45}{7} - \frac{10}{7} = \frac{35}{7} = 5 \quad \checkmark $$

The point $$\left(\frac{10}{7}, \frac{45}{7}\right)$$ satisfies $$y - x = 5$$.

The correct answer is Option 3: $$y - x = 5$$.