Question

The straight lines l₁ and l₂ pass through the origin and trisect the line segment of the line L: 9x + 5y = 45 between the axes. If m₁ and m₂ are the slopes of the lines l₁ and l_2, then the point of intersection of the line y = (m₁ + m₂)x with L lies on

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

Answer 1

The Correct Option is C

To solve this problem, we need to find the equation of the line that passes through the origin and trisects the line segment of the line \(L: 9x + 5y = 45\) between the axes.

First, determine the intercepts of the line \(L: 9x + 5y = 45\): 

• The x-intercept is obtained by setting \(y = 0\): \(9x = 45 \Rightarrow x = 5\). So, the x-intercept is (5, 0).
• The y-intercept is obtained by setting \(x = 0\): \(5y = 45 \Rightarrow y = 9\). So, the y-intercept is (0, 9).

The line segment between the axes is from (5, 0) to (0, 9). The task is to find points that trisect this segment.

Using section formula in the ratio 1:2 and 2:1:

• The point dividing (5, 0) to (0, 9) in the ratio 1:2: \(\left( \frac{1 \times 0 + 2 \times 5}{1 + 2}, \frac{1 \times 9 + 2 \times 0}{1 + 2} \right) = \left( \frac{10}{3}, 3 \right)\).
• The point dividing (5, 0) to (0, 9) in the ratio 2:1: \(\left( \frac{2 \times 0 + 1 \times 5}{2 + 1}, \frac{2 \times 9 + 1 \times 0}{2 + 1} \right) = \left( \frac{5}{3}, 6 \right)\).

The lines \(l_1\) and \(l_2\) are joining points (0, 0) to \(\left( \frac{10}{3}, 3 \right)\) and (0, 0) to \(\left( \frac{5}{3}, 6 \right)\) respectively.

The slopes are:

• For \(l_1\): \(m_1 = \frac{3}{\frac{10}{3}} = \frac{9}{10}\)
• For \(l_2\): \(m_2 = \frac{6}{\frac{5}{3}} = \frac{18}{5}\)

The equation of the line with slope \(m_1 + m_2\) becomes \(y = \left( \frac{9}{10} + \frac{18}{5} \right) x = \frac{45}{10}x = \frac{9}{2}x\).

We need to find where this line intersects with \(L: 9x + 5y = 45.\)

Substitute \(y = \frac{9}{2}x\) into \(9x + 5y = 45\):

\(9x + 5 \frac{9}{2}x = 45\)
\(9x + \frac{45}{2}x = 45\)
\(\frac{63}{2}x = 45\)
\(x = \frac{90}{63} = \frac{10}{7}\)
 

For \(y\):

\(y = \frac{9}{2} \times \frac{10}{7} = \frac{45}{7}\)

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

Now, check this point in each given line option to see if it satisfies:

For \(y-x=5\):

\(\frac{45}{7} - \frac{10}{7} = \frac{35}{7} = 5\) (True)

 

Thus, the point satisfies \(y-x=5\).

Therefore, the correct answer is y-x=5.