Question

The portion of the line \( 4x + 5y = 20 \) in the first quadrant is trisected by the lines \( L_1 \) and \( L_2 \) passing through the origin. The tangent of an angle between the lines \( L_1 \) and \( L_2 \) is:

• A. \(\frac{8}{5}\)
• B. \(\frac{25}{41}\)
• C. \(\frac{2}{5}\)
• D. \(\frac{30}{41}\)

Answer 1

The Correct Option is D

To solve this problem, we need to determine the tangent of the angle between the lines \(L_1\) and \(L_2\) which trisect the portion of the line \(4x + 5y = 20\) that lies in the first quadrant.

Step-by-step Solution 

1. First, find the intercepts of the line \(4x + 5y = 20\) in the first quadrant.
   • When \(y = 0\), \(4x = 20 \Rightarrow x = 5\) (x-intercept)
   • When \(x = 0\), \(5y = 20 \Rightarrow y = 4\) (y-intercept)
2. Determine the points that trisect this line segment.
   • Using section formula, the points that divide the segment in the ratio 1:2 and 2:1 are:
     • First point (Divide in 1:2 ratio): \[ \left(\frac{1 \times 0 + 2 \times 5}{1 + 2}, \frac{1 \times 4 + 2 \times 0}{1 + 2}\right) = \left(\frac{10}{3}, \frac{4}{3}\right) \]
     • Second point (Divide in 2:1 ratio): \[ \left(\frac{2 \times 0 + 1 \times 5}{2 + 1}, \frac{2 \times 4 + 1 \times 0}{2 + 1}\right) = \left(\frac{5}{3}, \frac{8}{3}\right) \]
3. Find the equations of the lines \(L_1\) and \(L_2\) that pass through these points from the origin.
   • Line \(L_1\) through \(\left(\frac{10}{3}, \frac{4}{3}\right)\):
     Equation: \(y = mx\), where \(m = \frac{4/3}{10/3} = \frac{2}{5}\).
   • Line \(L_2\) through \(\left(\frac{5}{3}, \frac{8}{3}\right)\):
     Equation: \(y = mx\), where \(m = \frac{8/3}{5/3} = \frac{8}{5}\).
4. Find the tangent of the angle \(\theta\) between these lines.
   • The formula for the tangent of the angle between two lines \(y = m_1x\) and \(y = m_2x\) is: \[ \tan \theta = \left|\frac{m_2 - m_1}{1 + m_1m_2}\right| \]
   • For \(m_1 = \frac{2}{5}\) and \(m_2 = \frac{8}{5}\), substitute into the formula: \[ \tan \theta = \left|\frac{\frac{8}{5} - \frac{2}{5}}{1 + \frac{2}{5} \times \frac{8}{5}}\right| = \left|\frac{\frac{6}{5}}{1 + \frac{16}{25}}\right| = \left|\frac{6/5}{41/25}\right| = \left|\frac{150/5}{41/5}\right| = \left|\frac{30}{41}\right| \]

Therefore, the tangent of the angle between the lines \(L_1\) and \(L_2\) is \(\frac{30}{41}\).

Answer 2

Step 1. Identify the Points Where the Line Intersects the Axes: The line \( 4x + 5y = 20 \) intersects the x-axis when \( y = 0 \):

 \(4x = 20 \Rightarrow x = 5\)
  
 So, the x-intercept is \( (5, 0) \). The line intersects the y-axis when \( x = 0 \):
  
 \(5y = 20 \Rightarrow y = 4\)
 
 So, the y-intercept is \( (0, 4) \).

Step 2. Determine the Coordinates of the Trisection Points: The line segment from \( (5, 0) \) to \( (0, 4) \) in the first quadrant is trisected at two points, dividing it into three equal parts. Using the section formula, the trisection points \( P \) and \( Q \) are:
\(P = \left( \frac{2 \cdot 0 + 1 \cdot 5}{3}, \frac{2 \cdot 4 + 1 \cdot 0}{3} \right) = \left( \frac{5}{3}, \frac{8}{3} \right)\)
 
\(Q = \left( \frac{1 \cdot 0 + 2 \cdot 5}{3}, \frac{1 \cdot 4 + 2 \cdot 0}{3} \right) = \left( \frac{10}{3}, \frac{4}{3} \right)\)

Step 3. Find the Slopes of Lines \( L_1 \) and \( L_2 \): 
  - Line \( L_1 \) passes through the origin and point \( P \left( \frac{5}{3}, \frac{8}{3} \right) \), so its slope \( m_1 \) is:
 
  \(m_1 = \frac{8/3}{5/3} = \frac{8}{5}\)

 - Line \( L_2 \) passes through the origin and point \( Q \left( \frac{10}{3}, \frac{4}{3} \right) \), so its slope \( m_2 \) is:

 \(m_2 = \frac{4/3}{10/3} = \frac{2}{5}\)

Step 4. Calculate the Tangent of the Angle Between \( L_1 \) and \( L_2 \): The tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:

\(\tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2}\)
 
  - Substituting \( m_1 = \frac{8}{5} \) and \( m_2 = \frac{2}{5} \):

  \(\tan \theta = \frac{\frac{8}{5} - \frac{2}{5}}{1 + \frac{8}{5} \cdot \frac{2}{5}} = \frac{\frac{6}{5}}{1 + \frac{16}{25}} = \frac{6/5}{41/25} = \frac{30}{41}\)

  Thus, the tangent of the angle between the lines \( L_1 \) and \( L_2 \) is \( \frac{30}{41} \).