1. Find the acute angle between the lines `x+3y+1=0` and `2x-y+4=0`Solution:We shall first find slopes of both the lines.
1. Slope of line `x+3y+1=0`
`x+3y+1=0`
`:. 3y=-x-1`
`:. y=-(x)/(3)-1/3`
`:. m_1=-1/3`
2. Slope of line `2x-y+4=0`
`2x-y+4=0`
`:. y=2x+4`
`:. m_2=2`
Let `theta` be the acute angle between the lines
then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`
`=|(-1/3-2)/(1+(-1/3)(2))|`
`=|(-7/3)/(1+-2/3)|`
`=|(-7/3)/(1/3)|`
`=7`
`:. theta=81.87^circ`
Hence, the acute angle between the given lines is `81.87^circ`
2. Find the acute angle between the lines `3x+2y+4=0` and `2x-3y-7=0`Solution:We shall first find slopes of both the lines.
1. Slope of line `3x+2y+4=0`
`3x+2y+4=0`
`:. 2y=-3x-4`
`:. y=-(3x)/(2)-2`
`:. m_1=-3/2`
2. Slope of line `2x-3y-7=0`
`2x-3y-7=0`
`:. 3y=2x-7`
`:. y=(2x)/(3)-7/3`
`:. m_2=2/3`
Let `theta` be the acute angle between the lines
then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`
`=|(-3/2-2/3)/(1+(-3/2)(2/3))|`
`=|(-13/6)/(1+-1)|`
`=|(-13/6)/(0)|`
`=1/0`
`:. theta=90^circ`
Hence, the acute angle between the given lines is `90^circ`
3. Find the acute angle between the lines `2x+3y+5=0` and `x-2y-4=0`Solution:We shall first find slopes of both the lines.
1. Slope of line `2x+3y+5=0`
`2x+3y+5=0`
`:. 3y=-2x-5`
`:. y=-(2x)/(3)-5/3`
`:. m_1=-2/3`
2. Slope of line `x-2y-4=0`
`x-2y-4=0`
`:. 2y=x-4`
`:. y=(x)/(2)-2`
`:. m_2=1/2`
Let `theta` be the acute angle between the lines
then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`
`=|(-2/3-1/2)/(1+(-2/3)(1/2))|`
`=|(-7/6)/(1+-1/3)|`
`=|(-7/6)/(2/3)|`
`=7/4`
`:. theta=60.26^circ`
Hence, the acute angle between the given lines is `60.26^circ`
4. Find the acute angle between the lines `3x-y+4=0` and `2x+y=3`Solution:We shall first find slopes of both the lines.
1. Slope of line `3x-y+4=0`
`3x-y+4=0`
`:. y=3x+4`
`:. m_1=3`
2. Slope of line `2x+y=3`
`2x+y=3`
`:. y=-2x+3`
`:. m_2=-2`
Let `theta` be the acute angle between the lines
then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`
`=|(3--2)/(1+(3)(-2))|`
`=|(5)/(1+-6)|`
`=|(5)/(-5)|`
`=1`
`:. theta=45^circ`
Hence, the acute angle between the given lines is `45^circ`
5. Find the acute angle between the lines `2x-y+3=0` and `x-3y+7=0`Solution:We shall first find slopes of both the lines.
1. Slope of line `2x-y+3=0`
`2x-y+3=0`
`:. y=2x+3`
`:. m_1=2`
2. Slope of line `x-3y+7=0`
`x-3y+7=0`
`:. 3y=x+7`
`:. y=(x)/(3)+7/3`
`:. m_2=1/3`
Let `theta` be the acute angle between the lines
then `tan theta=|(m_1-m_2)/(1+m_1m_2)|`
`=|(2-1/3)/(1+(2)(1/3))|`
`=|(5/3)/(1+2/3)|`
`=|(5/3)/(5/3)|`
`=1`
`:. theta=45^circ`
Hence, the acute angle between the given lines is `45^circ`
