Find the equation of the lines through the point (3, 2) which make an angle of 45^o with the line x – 2y = 3.

Let the slope of the required line be m₁.

The given line can be written as , which is of the from y = mx + c

Thus, Slope of the given line = m₂ =

It is given that the angle between the required line and line x – 2y = 3 is 45^o.

We know that if Ɵ is the acute angle between lines l₁ and l₂ with slopes m₁ and m₂ respectively, then

and

⇒ 2 + m₁ = 1- 2m₁ or 2 + m₁ = -1 + 2m₁

⇒ m₁ = or m₁ = 3

Now, when m₁ = 3, then,

The equation of the line passing through (3,2) and having a slope of 3 is:

y – 2 = 3(x – 3)

⇒ y -2 = 3x – 9

⇒ 3x – y = 7

And, when m₁ =

The equation of the line passing through (3,2) and having a slope of is:

3y – 6 = -x + 3

x + 3y = 9

Therefore, the equations of the lines are 3x – y = 7 and x + 3y = 9.