Question

Find equation of the line through the point (0, 2) making an angle with the positive x -axis. Also, find the equation of line parallel to it and crossing the y -axis at a distance of 2 units below the origin.

Answer 1

The line is passing through the point ( 0,2 ) making an angle of 2π 3 from the positive x- axis.

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 (1)

Substitute θ as 2π 3 in equation (1).

m=tan 2π 3 =tan( π− π 3 ) =−tan( π 3 ) =− 3

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 ) (2)

Substitute the values of ( x 0 , y 0 ) as ( 0,2 ) and slope m as − 3 in equation (2).

( y−2 )=− 3 ( x−0 ) y−2=− 3 x 3 x+y−2=0

Now, the line which is parallel to the line 3 x+y−2=0 have same slope of the line

It is given that the line is crossing the y-axis 2 units below the origin.

Thus, the coordinate of the point where the line cuts on y- axis is ( 0,−2 ) .

Substitute the value of ( x 0 , y 0 ) as ( 0,−2 ) and slope m as − 3 in equation (2).

( y−( −2 ) )=− 3 ( x−0 ) y+2=− 3 x 3 x+y+2=0

Thus, the equation of line is passing through the point ( 0,2 ) making an angle of 2π 3 from the positive x- axis is 3 x+y−2=0 and the equation of the line which is parallel to this line and crossing the y-axis 2 units below the origin is 3 x+y+2=0 .