Question

Find angles between the lines

Answer 1

The lines are 3 x+y=1 and x+ 3 y=1 .

The equation of line having slope m and making an intercept c with the y axis is given by

y=mx+c (1)

Let m 1 and m 2 be the slope of the lines 3 x+y=1 and x+ 3 y=1 .

Rearrange the terms of the equation of line 3 x+y=1 .

y=− 3 x+1

Compare the above equation with the equation (1).

m 1 =− 3

Rearrange the terms of the equation of line x+ 3 y=1 .

3 y=−x+1 y=− 1 3 x+ 1 3

Compare the above equation with the equation (1).

m 2 =− 1 3

The formula for the acute angle θ between two lines having slopes m 1 and m 2 is given by,

tanθ=| m 1 − m 2 1+ m 1 m 2 | (2)

Substitute the values of m 1 , m 2 as − 3 and − 1 3 in equation (2).

tanθ=| ( − 3 )−( − 1 3 ) 1+( − 3 )×( − 1 3 ) | =| − 3 + 1 3 1+1 | =| −3+1 3 2 | tanθ=| −2 2 3 |

Further simplify the above expression

tanθ=| −1 3 |

If the mod is opens up with the positive sign then,

θ= 1 3 =tan30° =30°

If the mod is opens up with the negative sign then,

θ=− 1 3 =−tan30° =tan( 180°−30° ) =tan( 150° )

Here θ=150° .

Thus, the angle between the lines 3 x+y=1 and x+ 3 y=1 are 30° or 150° .