Question

Show that the equation of the line passing through the origin and making an angle θ with the line .

Answer 1

The line passes through the origin. The line also makes an angle θ with the line y=mx+c.

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

y=mx+c(1)

Let the slope of the line passing through the origin be m 1 .

Substitute the value of m= m 1 and c=0 in equation (1).

y= m 1 x m 1 = y x

The formula for the acute angle 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 2 as m in equation (2).

tanθ=| m 1 −m 1+ m 1 ×m | =| y x −m 1+ y x m | =±( y x −m 1+ y x m )

If mod opens with positive sign.

tanθ=( y x −m 1+ y x m ) tanθ( 1+ y x m )=( y x −m ) tanθ+ y x m⋅tanθ= y x −m tanθ+ y x m⋅tanθ− y x +m=0

Further simplify the above expression.

y x ( m⋅tanθ−1 )+( m+tanθ )=0 y x =− ( m+tanθ ) ( m⋅tanθ−1 ) y x = ( m+tanθ ) ( 1−m⋅tanθ )

If mod opens with negative sign.

tanθ=−( y x −m 1+ y x m ) tanθ( 1+ y x m )=( m− y x ) tanθ+ y x m⋅tanθ=m− y x tanθ+ y x m⋅tanθ+ y x −m=0

Further simplify the above expression.

( tanθ−m )+ y x ( m⋅tanθ+1 )=0 y x ( m⋅tanθ+1 )=m−tanθ y x = m−tanθ ( m⋅tanθ+1 )

Thus, the equation of line passing through the origin and making an angle θ with the line y=mx+c is y x = ( m+tanθ ) ( 1−m⋅tanθ ) or y x = m−tanθ ( m⋅tanθ+1 ) .