Equations of the lines through the point (3, 2) and making an angle of 45° with line x – 2y – 3 = 0 are __________.

Let us suppose slope of line be m which makes an angle of 45^∘ with x − 2y − 3 = 0

$$\begin{gathered} \mathrm{Slope}\mathrm{of}x-2y-3=0\mathrm{is}\frac{1}{2} \\ \therefore \tan 45°=\frac{\left|m-{\displaystyle \frac{1}{2}}\right|}{\left|1+{\displaystyle \frac{m}{2}}\right|} \\ \mathrm{i}.\mathrm{e}.1=\frac{m-{\displaystyle \frac{1}{2}}}{1+{\displaystyle \frac{m}{2}}} \\ \mathrm{i}.\mathrm{e}.\pm \left(1+\frac{m}{2}\right)=m-\frac{1}{2} \\ \mathrm{i}.\mathrm{e}.\frac{m}{2}=\frac{3}{2}\mathrm{or}-1-\frac{m}{2}=m-\frac{1}{2} \\ \mathrm{i}.\mathrm{e}.m=3\mathrm{or}\frac{3m}{2}=-\frac{1}{2}\mathrm{i}.\mathrm{e}.m=-\frac{1}{3} \\ \therefore y=3x+c\mathrm{is}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{line}\mathrm{or}y=-\frac{1}{3}x+c \end{gathered}$$
Since this line passes through (3, 2)
i.e. 2 = 9 + c
i.e. c = – 7
∴ Equation of line is y = 3x – 7
i.e. 3x – y – 7 = 0

$$\begin{gathered} \mathrm{or}2=-\frac{1}{3}\left(3\right)+c \\ \mathrm{i}.\mathrm{e}.c=3 \\ \therefore y=-\frac{1}{3}x+3 \\ \mathrm{i}.\mathrm{e}.3y+x-9=0 \end{gathered}$$
∴ Equations of lines are 3x – y – 7 = or x + 3y – 9 = 0