Find the angle between each of the following pairs of straight lines:

(i) $3 x + y + 12 = 0$ and $x + 2 y - 1 = 0$

(ii) $3 x - y + 5 = 0$ and $x - 3 y + 1 = 0$

(iii) $3 x + 4 y - 7 = 0$ and $4 x - 3 y + 5 = 0$

(iv) $x - 4 y = 3$ and $6 x - y = 11$

(v) $(m^{2} - m n) y = (m n + n^{2}) x + n^{2}$ and $(m n + m^{2}) y = (m n + n^{2}) x + m^{3}$ .

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Solution

(i) The given equations are

$3 x + y + 12 = 0$

$x + 2 y - 1 = 0$

Writing the equation in the form

$y = m x + c$

$3 x + y + 12 = 0$

$y = - 3 x - 12$

$\Rightarrow m_{1} = - 3$

Also,

$x + 2 y - 1 = 0$

$2 y = 1 - x$

$y = \frac{1}{2} - \frac{x}{2}$

$\Rightarrow m_{2} = \frac{- 1}{2}$

Angle between the lines

$t a n θ = ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣$

$= ∣ ∣ ∣ ∣ \frac{- 3 - (\frac{- 1}{2})}{1 + (- 3) (\frac{- 1}{2})} ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ \frac{- 3 + \frac{1}{2}}{1 + \frac{3}{2}} ∣ ∣ ∣ = ∣ ∣ ∣ \frac{\frac{- 6 + 12}{2}}{\frac{2 + 3}{2}} ∣ ∣ ∣$

$= ∣ ∣ \frac{- 5}{5} ∣ ∣ = 1$

$\Rightarrow a n g l e = \frac{π}{4} = 45^{\circ}$

(ii) Finding slopes of the lines by converting the equation in the form

$y = m x + c$

$3 x - y + 5 = 0$

$\Rightarrow y = 3 x + 5$

$\Rightarrow m_{1} = 3$

Also,

$x - 3 y + 1 = 0$

$3 y = x + 1$

$y = \frac{x}{3} + \frac{1}{3}$

$\Rightarrow m_{2} = \frac{1}{3}$

Thus angle between the lines is

$t a n θ = ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣$

$= ∣ ∣ ∣ \frac{3 - \frac{1}{3}}{1 + 3 \frac{1}{3}} ∣ ∣ ∣ = ∣ ∣ ∣ \frac{\frac{9 - 1}{3}}{1 + 1} ∣ ∣ ∣$

$= ∣ ∣ ∣ \frac{\frac{8}{3}}{2} ∣ ∣ ∣ = ∣ ∣ \frac{8}{6} ∣ ∣ = \frac{4}{3}$

$\Rightarrow θ = t a n^{- 1} (\frac{4}{3})$

(iii) $3 x + 4 y - 7 = 0$ and $4 x - 3 y + 5 = 0$

To find angle between the lines, convert the equations in the form

$y = m x + c$

$3 x + 4 y - 7 = 0$

$\Rightarrow 4 y = - 3 x + 7$

$y = \frac{- 3}{4} x + \frac{7}{4}$

$m_{1} = \frac{- 3}{4}$

Also, $4 x - 3 y + 5 = 0$

$\Rightarrow 3 y = 4 x + 5$

$\Rightarrow y = \frac{4}{3} x + \frac{5}{3}$

$\Rightarrow m_{1} = \frac{4}{3}$

The angle between the line is given by

$t a n θ = ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣$

$= ∣ ∣ ∣ ∣ \frac{\frac{- 3}{4} - \frac{4}{3}}{1 + \frac{(- 3)}{4} (\frac{4}{3})} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ \frac{\frac{- 3}{4} - \frac{4}{3}}{1 - 1} ∣ ∣ ∣$

$\Rightarrow θ = \frac{π}{2}$ or $90^{\circ}$

(iv) $x - 4 y = 3$ and $6 x - y = 11$

To find angle convert the equation in the form $y = m x + c$

$x - 4 y = 3$

$\Rightarrow 4 y = x - 3$

$y = \frac{x}{4} - \frac{3}{4}$

$\Rightarrow m_{1} = \frac{1}{4}$

Also, $6 x - y = 11$

$y = 6 x - 11$

$\Rightarrow m_{2} = 6$

Thus, angle between the lines is

$t a n θ = ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣$

$= ∣ ∣ ∣ \frac{\frac{1}{4} - 6}{1 + \frac{1}{4} \times 6} ∣ ∣ ∣$

$= ∣ ∣ ∣ \frac{\frac{- 23}{4}}{1 + \frac{3}{2}} ∣ ∣ ∣ = ∣ ∣ ∣ \frac{\frac{- 23}{4}}{\frac{5}{2}} ∣ ∣ ∣$

$\Rightarrow θ = t a n^{- 1} (\frac{23}{10})$

(v) $(m^{2} - m n) y = (m n + n)^{2} x + n^{2}$ and $(n m + m^{2}) y = (m n + n^{2}) x + m^{3}$

Converting the equation in the form

$y = m x + c$

$y = \frac{(m n + n^{2})}{m^{2} - m n} x + \frac{n^{3}}{(m^{2} - m n)}$

$\Rightarrow m_{1} = \frac{m n + n^{2}}{m^{2} - m n}$

Also, $y = \frac{(m n - n^{2})}{n m + m^{2}} x + \frac{m^{3}}{n m + m^{2}}$

$\Rightarrow m_{2} = \frac{m n - n^{2}}{n m + m^{2}}$

Thus, angle between 2 lines is $t a n θ$

$\Rightarrow t a n θ = ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣$

$= ∣ ∣ ∣ ∣ \frac{(\frac{m n + n^{2}}{m^{2} - m n}) - (\frac{m n - n^{2}}{n m + m^{2}})}{1 + (\frac{m n + n^{2}}{m^{2} - m n}) (\frac{m n - n^{2}}{n m + m^{2}})} ∣ ∣ ∣ ∣$

$= ∣ ∣ \frac{m^{2} n^{2} + m^{3} n + n^{3} m + n^{2} m^{2} - m^{3} n + m^{2} n^{2} + n^{2} m^{2} - m n^{3}}{m^{3} n + m^{4} - m^{2} n^{2} - m^{3} n + m^{2} n^{2} - n m^{3} + m n^{3} - n^{4}} ∣ ∣$

$= ∣ ∣ \frac{4 m^{2} n^{2}}{m^{4} - n^{4}} ∣ ∣$

$\Rightarrow θ = t a n^{- 1} ∣ ∣ \frac{4 m^{2} n^{2}}{m^{4} - n^{4}} ∣ ∣$

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Find the angle between each of the following pairs of straight lines: (i) 3x+y+12=0 and x+2y−1=0 (ii) 3x−y+5=0 and x−3y+1=0 (iii) 3x+4y−7=0 and 4x−3y+5=0 (iv) x−4y=3 and 6x−y=11 (v) (m2−mn)y=(mn+n2)x+n2 and (mn+m2)y=(mn+n2)x+m3.