Question

Find the equation of two straight lines through the point (4,5) which makes an acute angle of 45 degree with 2x-y+7=0.

Answer 1

$2x-y+7=0$

$\Rightarrow y=2x+7$

$\therefore$ Slope of the required line $m_1=2$

$\Rightarrow \tan 45=\frac{\pm m_1-m}{1+m_{1}m}=\frac{\pm 2-m}{1+2m}$

$\Rightarrow 1=\pm \frac{2-m}{1+2m}$

Take positive sign $1+2m=2-m$

$\Rightarrow 3m=1$

$\therefore m=\frac{1}{3}$

Take negative sign $1+2m=-2+m$

$\Rightarrow m=-3$

Equation of the straight line passing through point $(4,5)$ and its slope $=m$ is $y-5=m(x-4)$

when $m=\frac{1}{3}$ equation of straight line

$\Rightarrow y-5=\frac{1}{3}(x-4)$

$\Rightarrow 3y-15=x-4$

$\Rightarrow x-3y+11=0$

and when $m=-3$ $y-5=-3(x-4)$

$\Rightarrow 3x+y-17=0$

Hence, solved.