Question

Find the equations of the straight lines passing through the point $(-3,2)$ and making an angle of ${45}^{th}$ with the straight line $3x-y+4=0$.

Answer 1

Let slope of required line be $m_1$

$y=3x+4$ is of form $y=mx+c$ and has slope $m_2=3$

$\Rightarrow \tan 45=\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|=\frac{30m_1}{1+3m}=1$

$\Rightarrow 3-m_1=1+3m$,

or $3-m_1=-1-3m_1$$

$\Rightarrow 4m_1=2\Rightarrow m_1=\frac{1}{2}$ or $2m_1=-4$

$\Rightarrow m_1=-2$

$\Rightarrow$ Equations of straight lines

$y-2=\frac{1}{2}(x+3)$ or $y-2=-2(x+5)$

$\Rightarrow 2y-4=x+3$ $\Rightarrow y-2=-2x-6$

$\Rightarrow 2y=x+7$ $y=-2x-4$.