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

Q: 11     Find the equation of the lines through the point \small (3,2) which make an angle of  \small 45^{\circ} with the line  \small x-2y=3

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Given the equation of the line is
\small x-2y=3
The slope of line \small x-2y=3 , m_2= \frac{1}{2}
Let the slope of the other line is, m_1=m
Now, it is given that both the lines make an angle \small 45^{\circ} with each other 
Therefore,
\tan \theta = \left | \frac{m_2-m_1}{1+m_1m_2} \right |
\tan 45\degree = \left | \frac{\frac{1}{2}-m}{1+\frac{m}{2}} \right |
1= \left | \frac{1-2m}{2+m} \right |
Now,

Case (i)
1=\frac{1-2m}{2+m}
2+m=1-2m
m = -\frac{1}{3}                      
Equation of line passing through the point  \small (3,2)  and  with slope -\frac{1}{3}
(y-2)=-\frac{1}{3}(x-3)
3(y-2)=-1(x-3)
x+3y=9 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)

Case (ii)
1=-\left ( \frac{1-2m}{2+m} \right )
2+m=-(1-2m)
m= 3
Equation of line passing through the point  \small (3,2)  and  with slope 3  is
(y-2)=3(x-3)
3x-y=7 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)

Therefore, equations of lines are 3x-y=7  and x+3y=9

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