NCERT Solutions
Class 11 Maths
Straight Lines
Ex.Misc. Q.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.
Given equation of line are:
y = 3x + 1 .............. (1)
Slope = 3
2y = x + 3
=> y =
+ 
........... (2)
Slope = 
y = mx + 4 ........ (3)
Slope = m
Given that equation (1) and (2) are equally inclined with equation (3).
So, the angle between equation (1) and (3) = angle between equation (2) and (3)
=> | (3 - m) ÷ (1 + 3m) | = | (
- m) ÷ (1 + 
)|
=> | (3 - m) ÷ (1 + 3m) | = | (1 - 2m) ÷ (2 + m) |
=> {(3 - m) ÷ (1 + 3m)} = {(1 - 2m) ÷ (2 + m)} and {(3 - m) ÷ (1 + 3m)} = {-(1 - 2m) ÷ (2 + m)}
=> (3 - m) (2 + m) = (1 - 2m) (1 + 3m) and (3 - m) × (2 + m) = -(1 + 3m) (1 - 2m)
=> 6 + 3m - 2m - m2 = 1 + 3m - 2m - 6 m2 and 6 + 3m - 2m - m2 = -(1 + 3m - 2m - 6 m2)
=> -m2 + m + 6 = 1 + m - 6m2 and -m2 + m + 6 = -1 – m + 6m2
=> 5m2 + 5 = 0 and 7m2 - 2m - 7 = 0
=> m2 + 1= 0
=> m2 = -1 which is not possible.
Again, 7m2 - 2m - 7 = 0
=> m = {2 + √ (4 – 4 × 7 × (-7))} ÷ (2 × 7) and m = {2 - √ (4 – 4 × 7 × (-7))} ÷ (2 × 7)
=> m = {2 + √ (4 + 4 × 49)} ÷ 14 and m = {2 - √ (4 + 4 × 49)} ÷ 14
=> m = {2 + 2√ (1 + 49)} ÷ 14 and m = {2 - 2√ (1 + 49)} ÷ 14
=> m = {1 + √50} ÷ 7 and m = {1 - √50} ÷ 7
=> m = {1 + 5√2} ÷ 7 and m = {1 - 5√2} ÷ 7
Hence, the value of m is {1 ± 5√2} ÷ 7