Question

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

Answer 1

Equation of line AB, Slope of y = 3x + 1 = 3

Equation of line BC, y = mx + 4, slope = m

If there is an angle θ between them, then

`tan θ = ("m" - 3)/(1 + 3"m")`    ..........(i)

Equation of line AC, 2y = x + 3

or y = `1/2 "x" + 3/2`

Slope of AC = `1/2`

When the angle between AB and AC is θ, then

`tan θ = ± ("m" - 1/2)/(1/2"m") = ±(2"m" - 1)/(2 + "m")`     .......(ii)

From equation (i) and equation (ii),

`("m" - 3)/(1 + 3"m") = ± (2"m" - 1)/(2 + "m")`

with +ve sign, `("m" - 3)/(1 + 3"m") = ± (2"m" - 1)/(2 + "m")`

∴ (2m − 1)(3m + 1) = (m + 2)(m − 3)

or  6m² − m − 1 = m² − m − 6

∴ m² = −1 is not valid.

with -ve sign, `("m" - 3)/(1 + 3"m") = -(2"m" - 1)/(2 + "m")`

(3m + 1)(2m − 1) + (m + 3)(m + 2) = 0

or (6m² − m − 1) + (m² − m − 6) = 0

or 7m² − 2m − 7 = 0

∴ m = `(2 ± sqrt(4 + 4 xx 49))/14`

= `(2 ± sqrt(200))/14`

= `(2 ± 10sqrt2)/14`

= `(1 ± 5sqrt2)/7`

Hence, required value of m = `(1 ± 5sqrt2)/7`.