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

Let the slope of the required line be *m*_{1}.

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

∴Slope of the given line =

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

We know that if *θ*isthe acute angle between lines *l*_{1} and *l*_{2} with slopes *m*_{1} and *m*_{2 }respectively, then.

**Case I:** *m*_{1} = 3

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

*y* – 2 = 3 (*x *– 3)

*y* – 2 = 3*x* – 9

3*x* – *y* = 7

**Case II:** *m*_{1} =

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

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

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