Question

Find the equation of the line which passes though and is inclined with the x -axis at an angle of 75°.

Answer 1

The point through which the line passes is ( 2,2 3 ). The line is inclined at an angle of 75° to the x-axis.

The formula for the equation of a non-vertical line having slope m and passing through the point ( x 0 , y 0 ) is given by,

( y− y 0 )=m( x− x 0 )(1)

Substitute the values of ( x 0 , y 0 ) as ( 2,2 3 ) in equation (1).

( y−2 3 )=m( x−2 )(2)

The formula for the slope m of a line subtends an angle θ in anticlockwise direction with positive x-axis is given by,

m=tanθ =tan75° =tan( 45°+30° ) = tan45°+tan30° 1−tan45°⋅tan30°

Further simplify the above relation.

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

Substitute the values of m from equation (3) to equation (2).

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

Further simplify the above expression,

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

Thus the equation of line passing through the point ( 2,2 3 ) and inclined at angle of 75° is ( 3 +1 )x−( 3 −1 )y=4( 3 −1 ).