Find the tangent of the angle between the lines $2 x + 3 y - 8 = 0$ and $5 x - y + 7 = 0$ .

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Solution

Given two lines $L_{1}, L_{2}$ ,where
$L_{1} := 2 x + 3 y - 8 = 0 \Rightarrow y = \frac{- 2}{3} x - \frac{8}{3} L_{2} := 5 x - y + 7 = 0 \Rightarrow y = 5 x + 7$
the slopes of line $L_{1}$ & $L_{2}$ be $m_{1}$ & $m_{1}$ respectively then $m_{1} = \frac{- 2}{3} m_{2} = 5$
$∴ tan θ =$ tangent of the angle between the lines
$= ∣ ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣ ∣ (m_{1} m_{2} \neq - 1) = ∣ ∣ ∣ ∣ ∣ ∣ \frac{\frac{- 2}{3} - 5}{1 - \frac{10}{3}} ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ \frac{\frac{- 17}{15}}{\frac{- 7}{3}} ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ \frac{17}{35} ∣ ∣ ∣ ∴ tan θ = \frac{17}{35}$

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