Q. 114.2( 41 Votes )

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

Answer :

Let the slope of the required line be m_{1}.

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

Thus, Slope of the given line = m_{2} =

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 l_{1} and l_{2} with slopes m_{1} and m_{2} respectively, then

and

⇒ 2 + m_{1} = 1- 2m_{1} or 2 + m_{1} = -1 + 2m_{1}

⇒ m_{1} = or m_{1} = 3

Now, when m_{1} = 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 m_{1} =

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.

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