Question

Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of one line is 2, find equation of the other line.

Answer 1

The two lines are passes through the point ( 2,3 ) and intersect each other at an angle of 60° .

The slope of one of the line is 2 .

Let the slope of another line is m 2 .

The formula for the 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 | (1)

Substitute the values of θ m 1 , m 2 as 60° , 2 and m 2 in equation (1).

tan60°=| 2− m 2 1+2× m 2 | 3 =| 2− m 2 1+2 m 2 | 3 =±( 2− m 2 1+2 m 2 )

If the mod opens with the positive sign then,

3 = 2− m 2 1+2 m 2 3 ⋅( 1+2 m 2 )=2− m 2 3 +2 3 m 2 =2− m 2 2 3 m 2 + m 2 =2− 3

Further simplify the above equation.

( 2 3 +1 ) m 2 =2− 3 m 2 = 2− 3 ( 2 3 +1 )

The formula for the equation of the line having slope m passes through the point ( x 1 , y 1 ) is given by,

( y− y 1 )=m( x− x 1 ) (2)

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

( y−3 )= 2− 3 ( 2 3 +1 ) ⋅( x−2 ) ( 2 3 +1 )⋅( y−3 )=( 2− 3 )⋅( x−2 ) ( 2 3 +1 )⋅y−6 3 −3=( 2− 3 )⋅x−4+2 3 ( 2 3 +1 )y=( 2− 3 )x−4+2 3 +6 3 +3

Further simplify the above expression.

( 2 3 +1 )y=( 2− 3 )x−1+8 3 −( 2− 3 )x+( 2 3 +1 )y=−1+8 3 ( 3 −2 )x+( 2 3 +1 )y=−1+8 3

If the mod opens with the negative sign then,

3 =−( 2− m 2 1+2 m 2 ) 3 ⋅( 1+2 m 2 )=−( 2− m 2 ) 3 +2 3 m 2 = m 2 −2 2 3 m 2 − m 2 =−2− 3

Further simplify the above equation.

( 2 3 −1 ) m 2 =−( 2+ 3 ) m 2 = −( 2+ 3 ) ( 2 3 −1 )

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

( y−3 )= −( 2+ 3 ) ( 2 3 −1 ) ⋅( x−2 ) ( 2 3 −1 )⋅( y−3 )=−( 2+ 3 )⋅( x−2 ) ( 2 3 −1 )⋅y−6 3 +3=−( 2+ 3 )⋅x+4+2 3 ( 2 3 +1 )y=−( 2+ 3 )x+4+2 3 +6 3 −3

Further simplify the above expression.

( 2 3 +1 )y=−( 2+ 3 )x+1+8 3 ( 2+ 3 )x+( 2 3 +1 )y=1+8 3

Thus the equation of two lines passes through the point ( 2,3 ) and intersect each other at an angle of 60° is ( 3 −2 )x+( 2 3 +1 )y=−1+8 3 and ( 2+ 3 )x+( 2 3 +1 )y=1+8 3 .