Question

Question 19 If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

Class X1 - Maths -Straight Lines Page 234

Answer 1

Let angle between two lines y = m1x + c1 and y =m2x +c2 is ∅.
then tan∅ = |m1 -m2|/|1 + m1.m2|

solution :-
Given,
y = 3x + 1 , slope of it {m1} = 3
2y = x + 3 , slope of it {m2} = 1/2

angle between y = 3x + 1 and y = mx + 4 is ∅
then , tan∅ = |3 - m|/| 1 + 3m| ____(1)

it is given that angle between the lines 2y = x + 3 and y = mx + 4 is also ∅ .
then, tan∅ = |1/2 - m|/| 1 +m/2|________(2)

from equations (1) and (2),
|3 - m|/|1 + 3m| = | 1/2 - m|/| 1 + m/2|
|3 - m|/|1 + 3m| = |1 - 2m|/|2 + m|

taking positive sign ,
(3 - m)(2 + m) = (1 - 2m)( 1 + 3m)
6 + 3m -2m -m² = 1 + 3m -2m - 6m²
5m² = -5
m² = -1 it's not possible .

taking negative sign,
(3 - m)(2 + m) = -(1 - 2m)(1 + 3m)
6 + 3m - 2m - m² = -1 - 3m + 2m + 6m²
7m² - 2m -7 = 0
m = { 2 ± √(4 + 49 × 4)}/14
= { 2 ± 2√50}/14
= { 1 ± √50}/7
= {1 ± 5√2}/7

hence, m = { 1 ± 5√2}/7

Answer 2

Let angle between two lines y = m1x + c1 and y =m2x +c2 is ∅.

then tan∅ = |m1 -m2|/|1 + m1.m2|

solution :-

Given,

y = 3x + 1 , slope of it {m1} = 3

2y = x + 3 , slope of it {m2} = 1/2

angle between y = 3x + 1 and ybrain 4 is ∅

then , tan∅ = |3 - m|/| 1 + 3m| ____(1)

it is given that angle between the lines 2y = x + 3 and y = mx + 4 is also ∅ .

then, tan∅ = |1/2 - m|/| 1 +m/2|________(2)

from equations (1) and (2),

|3 - m|/|1 + 3m| = | 1/2 - m|/| 1 + m/2|

|3 - m|/|1 + 3m| = |1 - 2m|/|2 + m|

taking positive sign ,

(3 - m)(2 + m) = (1 - 2m)( 1 + 3m)

6 + 3m -2m -m² = 1 + 3m -2m - 6m²

5m² = -5

m² = -1 it's not possible .

taking negative sign,

(3 - m)(2 + m) = -(1 - 2m)(1 + 3m)

6 + 3m - 2m - m² = -1 - 3m + 2m + 6m²

7m² - 2m -7 = 0

m = { 2 ± √(4 + 49 × 4)}/14

= { 2 ± 2√50}/14

= { 1 ± √50}/7

= {1 ± 5√2}/7

hence, m = { 1 ± 5√2}/7