Question

The slope of a line is double of the slope of another line. If tangent of the angle between them is , find the slopes of he lines.

Answer 1

The slope of first line is double the slope of another line. The tangent of the angle between the two lines is 1 3 .

Let m 1 and m 2 be the slope of the two lines.

Let θ be the angle between the two lines.

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

tanθ=| m 2 − m 1 1+ m 1 m 2 |(1)

Substitute the value of tanθ as 1 3 in equation (1).

1 3 =| m 2 − m 1 1+ m 1 m 2 |

According to the given condition,

m 1 =2 m 2

Substitute the value of m 1 in the above expression.

1 3 =| m 2 −2 m 2 1+2 m 2 m 2 | 1 3 =| − m 2 1+2 m 2 2 |

If mod opens with the positive sign; then,

1 3 = − m 2 1+2 m 2 2 −3 m 2 =1+2 m 2 2 2 m 2 2 +3 m 2 +1=0 2 m 2 2 +2 m 2 + m 2 +1=0

Further simplify the above expression.

2 m 2 ( m 2 +1 )+1( m 2 +1 )=0 ( 2 m 2 +1 )( m 2 +1 )=0

Equate the coefficients on both the sides.

m 2 =− 1 2  or -1

If m 2 =−1

m 1 =2 m 2 =2×( −1 ) =−2

If m 2 =−1; then, slope of the lines are −1 and −2.

If m 2 = −1 2

m 1 =2 m 2 =2×( −1 2 ) =−1

If m 2 =− 1 2 ; then, slope of the lines are −1 and − 1 2 .

If mod opens with the negative sign; then,

1 3 = m 2 1+2 m 2 2 3 m 2 =1+2 m 2 2 2 m 2 2 −3 m 2 +1=0 2 m 2 2 −2 m 2 − m 2 +1=0

Further simplify the above expression.

2 m 2 ( m 2 −1 )−1( m 2 +1 )=0 ( 2 m 2 −1 )( m 2 −1 )=0

Equate the coefficients on both the sides.

m 2 = 1 2  or 1

If m 2 =1

m 1 =2 m 2 =2×1 =2

If m 2 =1; then, slope of the lines are 1 and 2.

If m 2 = 1 2

m 1 =2 m 2 =2× 1 2 =1

If m 2 =1; then, slope of the lines are 1 and 2.

Thus, the slope of the pair of lines are either −1,−2 and − 1 2 ,−1 or 1,2 and 1 2 ,1.