Question

15. Find the equations of the straight lines passing through the point (-3, 2) and making
an angle of 45° with the straight line 3x - y + 4 = 0.​

Answer 1

EXPLANATION.

=> equation of straight lines passing through

the point (-3,2)

=> making an angle of = 45°

=> straight line = 3x - y + 4 = 0

=> Slope of the equation = -a/b

=> 3x - y + 4 = 0 => -3/-1 = 3 = slope

$\sf : \implies \: | \dfrac{ m_{1} - m_{2}}{1 + m_{1} m_{2} } | = \tan(45 \degree) \\ \\ \sf : \implies \: | \frac{ m_{1} - 3 }{1 + 3 m_{1}} | = 1 \\ \\ \sf : \implies \: \pm \: ( \frac{m_{1} \: - 3 }{1 + 3 m_{1} } ) = 1$

$\sf : \implies \: m_{1} - 3 = 1 + 3 m_{1} \\ \\ \sf : \implies \: - 3 - 1 = 3 m_{1} \: - m_{1} \\ \\ \sf : \implies \: - 4 = 2 m_{1} \\ \\ \sf : \implies \: - 2 = m_{1} \: \: \: .....(1)$

$\sf : \implies \: - ( m_{1} - 3) = 1 + 3 m_{1} \\ \\ \sf : \implies \: - m_{1} + 3 = 1 + 3m_{1} \\ \\ \sf : \implies \: 3 - 1 = 3 m_{1} + m_{1} \\ \\ \sf : \implies \: 2 = 4 m_{1} \\ \\ \sf : \implies \: m_{1} \: = \frac{1}{2} \: \: \: .....(2)$

=> Line Passing through point = (-3,2)

$\sf : \implies \: equation \: of \: line \implies \: (y - y_{1}) = m(x - x_{1}) \\ \\ \sf : \implies \: (y - (2)) = - 2(x - ( - 3)) \\ \\ \sf : \implies \: (y - 2) = - 2(x + 3) \\ \\ \sf : \implies \: y - 2 = - 2x - 6 \\ \\ \sf : \implies \: y + 2x = - 4$

$\sf : \implies \: (y - 2) = \dfrac{1}{2} (x - ( - 3)) \\ \\ \sf : \implies \: (y - 2) = \frac{1}{2} (x + 3) \\ \\ \sf : \implies \: 2(y - 2) = (x + 3) \\ \\ \sf : \implies \: 2y - 4 = x + 3 \\ \\ \sf : \implies \: 2y - x = 7$