Question

If the lines y = 3 x + 1 and 2 y = x + 3 are equally inclined to the line y = mx + 4, find the value of m .

Answer 1

The lines y=3x+1 and 2y=x+3 are equally inclined to the line y=mx+4.

The equation of line with slope m and intercept on y axis as c is given by,

y=mx+c(1)

Let m 1 , m 2 and m 3 be the slopes of three lines.

Compare the equation of line with equation (1).

m 1 =3,  m 2 = 1 2 ,  m 3 =m

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)

Let θ 1 and θ 2 be the angles made by the lines y=3x+1 and 2y=x+3 with the line y=mx+4 respectively.

Substitute the values of m 1 and m 2 as 3,m respectively in equation (1).

tan θ 1 =| 3−m 1+3m |

Substitute the value of m 1 and m 2 as 1 2 ,m respectively in equation (1).

tan θ 2 =| 1 2 −m 1+ 1 2 m |

As the lines y=3x+1 and 2y=x+3 are equally inclined to the line y=mx+4,

tan θ 1 =tan θ 2 (3)

Substitute the values of tan θ 1 and tan θ 2 in equation (3).

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

If the mod opens with the positive sign; then,

3−m 1+3m = 1−2m 2+m ( 2+m )⋅( 3−m )=( 1−2m )⋅( 1+3m ) 6+3m−2m− m 2 =1+3m−2m−6 m 2 6+m− m 2 =1+m−6 m 2

Further simplify the above expression.

5 m 2 +5=0 5( m 2 +1 )=0 m 2 =−1 m= −1

The value of m is not real, so not possible.

If the mod opens with the negative sign; then,

3−m 1+3m =−( 1−2m 2+m ) ( 2+m )⋅( 3−m )=( 2m−1 )⋅( 1+3m ) 6+3m−2m− m 2 =2m−1+6 m 2 −3m 6+m− m 2 =6 m 2 −m−1

Further simplify the above expression.

7 m 2 −2m−7=0 m= 2± 4−4( 7 )( −7 ) 2( 7 ) m= 2±2 1+49 2( 7 ) m= 1± 50 7

Further simplify the above expression.

m= 1±5 2 7

Thus, the value of m for which the lines y=3x+1 and 2y=x+3 are equally inclined to the line y=mx+4 is 1±5 2 7 .