© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Find the intercepts of the given line
The equation of the line is
$4x + 5y = 20$.
• To find the x-intercept, set $y = 0$, giving $x = 5$.
• To find the y-intercept, set $x = 0$, giving $y = 4$.
Thus, the line meets the coordinate axes at $(5,0)$ and $(0,4)$.
Step 2: Trisect the line segment in the first quadrant
We want to trisect the segment $PQ$, where $P = (5,0)$ and $Q = (0,4)$.
The vector $Q - P = (-5,4)$.
For trisection points $R_1$ and $R_2$ that divide the segment into three equal parts, we use the concept of section formula or vector partition:
$R_1 = P + \dfrac{1}{3}(Q - P)
= (5,0) + \dfrac{1}{3}\bigl(-5,4\bigr)
= \Bigl(5 - \dfrac{5}{3},\,\dfrac{4}{3}\Bigr)
= \Bigl(\dfrac{10}{3},\,\dfrac{4}{3}\Bigr).$
$R_2 = P + \dfrac{2}{3}(Q - P)
= (5,0) + \dfrac{2}{3}\bigl(-5,4\bigr)
= \Bigl(5 - \dfrac{10}{3},\,\dfrac{8}{3}\Bigr)
= \Bigl(\dfrac{5}{3},\,\dfrac{8}{3}\Bigr).$
Step 3: Determine slopes of lines from the origin
The lines $L_1$ and $L_2$ pass through the origin $(0,0)$ and the points
$R_1\bigl(\tfrac{10}{3},\,\tfrac{4}{3}\bigr)$ and
$R_2\bigl(\tfrac{5}{3},\,\tfrac{8}{3}\bigr)$ respectively.
Hence, the slopes $m_1$ and $m_2$ are:
$m_1 = \dfrac{\tfrac{4}{3}}{\tfrac{10}{3}} = \dfrac{4}{10} = \dfrac{2}{5}.$
$m_2 = \dfrac{\tfrac{8}{3}}{\tfrac{5}{3}} = \dfrac{8}{5}.$
Step 4: Calculate the tangent of the angle between the lines
If two lines have slopes $m_1$ and $m_2$, the tangent of the angle $\theta$ between them is given by:
$$
\tan(\theta) = \left|\dfrac{m_1 - m_2}{1 + m_1 \, m_2}\right|.
$$
Plug in $m_1 = \dfrac{2}{5}$ and $m_2 = \dfrac{8}{5}$:
$$
\tan(\theta)
= \left|\dfrac{\tfrac{2}{5} - \tfrac{8}{5}}{1 + \left(\tfrac{2}{5}\right)\left(\tfrac{8}{5}\right)}\right|
= \left|\dfrac{-\tfrac{6}{5}}{1 + \tfrac{16}{25}}\right|
= \dfrac{\tfrac{6}{5}}{\tfrac{25}{25} + \tfrac{16}{25}}
= \dfrac{\tfrac{6}{5}}{\tfrac{41}{25}}
= \dfrac{6}{5} \times \dfrac{25}{41}
= \dfrac{30}{41}.
$$
Final Answer
The tangent of the angle between the lines $L_1$ and $L_2$ is
$\dfrac{30}{41}$.
