The equation of the lines passing through the point $(3,2)$ and making an acute angle of $45^{\circ}$ with…
The equation of the lines passing through the point $(3,2)$ and making an acute angle of $45^{\circ}$ with the lien $x-2 y-3=0$ are
$x+2 y-7=0,2 x-y-4=0$
$3 x+y-11=0, x+3 y-9=0$
$3 x-y-7=0, x+3 y-9=0$
$3 x+y-11=0, x+3 y+9=0$
Solution
Let the slope of required line by $\mathrm{m}$ then $\tan 45^{\circ}=\left|\frac{m-\frac{1}{2}}{1+m \times \frac{1}{2}}\right|$
$\begin{aligned} & \Rightarrow \pm 1=\frac{m-\frac{1}{2}}{1+m \times \frac{1}{2}} \\ & \Rightarrow m=3 \text { or } m=-\frac{1}{3}\end{aligned}$
So, required equation
$\begin{aligned} & (y-2)=3(x-3) \text { or }(y-2)=\frac{-1}{3}(x-3) \\ & \Rightarrow 3 x-y-7=0 \text { or } x+3 y-9=0\end{aligned}$