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Question

The equation of the bisector of the acute angle between the lines $$\displaystyle 2x+  y + 4 = 0$$ and $$\displaystyle x +2y = 1$$ is


A
x+y+5=0
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B
x+y+1=0
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C
xy=5
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D
x+y=5
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Solution

The correct option is A $$\displaystyle x+y+1=0$$
The angle of bisector of $$2x+y+4=0$$ and $$x+2y=1$$ is 
$$\displaystyle \frac { 2x+y+4 }{ \sqrt { 4+1+16 }  } =\pm \frac { x+2y-1 }{ \sqrt { 1+4+1 }  } \Rightarrow \frac { 2x+y+4 }{ \sqrt { 21 }  } =\pm \frac { x+2y+1 }{ \sqrt { 6 }  } $$
Now for $$2x+y+4=0$$ and $$-x-2y+1=0$$
$${ a }_{ 1 }{ a }_{ 2 }+{ b }_{ 1 }{ b }_{ 2 }=2\times \left( -1 \right) +1\times \left( -2 \right) <0$$
Therefore equation of angle bisector of acute angle is with negative sign
$$\displaystyle \frac { 2x+y+4 }{ \sqrt { 21 }  } =-\frac { x+2y+1 }{ \sqrt { 6 }  } \Rightarrow x+y+1=0$$

Mathematics

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