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

Options

A. y – 2x = 5
B. 6x + y = 10
C. y – x = 5
D. 6x – y = 15

Answer

To solve the problem step by step, we will follow the outlined approach:

### Step 1: Find the intercepts of the line \( L: 9x + 5y = 45 \)

To find the x-intercept, set \( y = 0 \):
\[ 9x + 5(0) = 45 \implies 9x = 45 \implies x = 5 \]
So, the x-intercept is \( A(5, 0) \).

To find the y-intercept, set \( x = 0 \):
\[ 9(0) + 5y = 45 \implies 5y = 45 \implies y = 9 \]
So, the y-intercept is \( B(0, 9) \).

### Step 2: Determine the coordinates of the trisection points

The line segment \( AB \) can be trisected into three equal parts. The coordinates of the trisection points can be calculated as follows:

1. The first trisection point \( P \) divides \( AB \) in the ratio \( 1:2 \):
\[ P = \left( \frac{1 \cdot 0 + 2 \cdot 5}{1 + 2}, \frac{1 \cdot 9 + 2 \cdot 0}{1 + 2} \right) = \left( \frac{10}{3}, 3 \right) \]

2. The second trisection point \( Q \) divides \( AB \) in the ratio \( 2:1 \):
\[ Q = \left( \frac{2 \cdot 0 + 1 \cdot 5}{2 + 1}, \frac{2 \cdot 9 + 1 \cdot 0}{2 + 1} \right) = \left( \frac{5}{3}, 6 \right) \]

### Step 3: Calculate the slopes \( m_1 \) and \( m_2 \)

The slopes of the lines \( l_1 \) and \( l_2 \) passing through the origin and the points \( P \) and \( Q \) respectively are calculated as follows:

1. For line \( l_1 \) through \( P \):
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{\frac{10}{3} - 0} = \frac{3}{\frac{10}{3}} = \frac{9}{10} \]

2. For line \( l_2 \) through \( Q \):
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 0}{\frac{5}{3} - 0} = \frac{6}{\frac{5}{3}} = \frac{18}{5} \]

### Step 4: Find \( m_1 + m_2 \)

Now, we calculate \( m_1 + m_2 \):
\[ m_1 + m_2 = \frac{9}{10} + \frac{18}{5} = \frac{9}{10} + \frac{36}{10} = \frac{45}{10} = \frac{9}{2} \]

### Step 5: Write the equation of the line \( y = (m_1 + m_2)x \)

The equation of the line becomes:
\[ y = \frac{9}{2}x \]

### Step 6: Find the intersection of this line with line \( L \)

Substituting \( y = \frac{9}{2}x \) into the equation of line \( L \):
\[ 9x + 5\left(\frac{9}{2}x\right) = 45 \]
This simplifies to:
\[ 9x + \frac{45}{2}x = 45 \implies \left(9 + \frac{45}{2}\right)x = 45 \]
Converting \( 9 \) to a fraction:
\[ \frac{18}{2} + \frac{45}{2} = \frac{63}{2} \implies \frac{63}{2}x = 45 \]
Thus,
\[ x = \frac{45 \cdot 2}{63} = \frac{90}{63} = \frac{10}{7} \]

### Step 7: Find \( y \)

Now substituting \( x = \frac{10}{7} \) back into \( y = \frac{9}{2}x \):
\[ y = \frac{9}{2} \cdot \frac{10}{7} = \frac{90}{14} = \frac{45}{7} \]

### Step 8: Conclusion

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