Question

Find the equation of the lines through the point (3, 2) which make an angle of ${45}^{\circ }$ with the line x - 2y = 3

Answer 1

Let m be the slope of required line which passes through point (3, 2). Then equation of required line is

y - 2 = m(x - 3) ...(i)

The equation of given line is x - 2y = 3

$\Rightarrow y=\frac{x}{2}-\frac{3}{2}$ . . . (ii)

$\therefore$ Slope of given line is $\frac{1}{2}$

It is given that lines (i) and (it) make an angle of ${45}^{\circ }$.

$\therefore tan{45}^{\circ }=\left|\frac{m-\frac{1}{2}}{1+\frac{m}{2}}\right|$

$\Rightarrow 1=\left|\frac{2m-1}{2+m}\right|$

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

When $\frac{2m-1}{2+m}=1$

$\Rightarrow 2m-1=2+m\Rightarrow m=3$

Then equation of required line is

y - 2 = 3(x - 3).

$\Rightarrow$ y - 2 = 3x - 9 $\Rightarrow$ 3x - y - 7 = 0

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

$\Rightarrow 3m=-1\Rightarrow m=\frac{-1}{3}$

Then equation of required line is

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

$\Rightarrow 3y-6=-x+3\Rightarrow x+3y=9=0$