If the line, 2 x - y + 3 = 0 is at a distance 1 5 and 2 5 from the lines 4 x - 2 y + α = 0 and 6 x - 3 y + β = 0 respectively, then the sum of all possible values of α and β is ____________.

Step-by-step solution

2   x - y + 3 = 0 … … ( i ) 4 x - 2 y + α = 0    ⇒    2 x - y + α 2 = 0 … … ( i i ) 6 x - 3 y + β = 0    ⇒    2 x - y + β 3 = 0 … … ( i i i ) d 1 = α 2 - 3 2 2 + 1 2 = 1 5 ⇒ | α - 6 | = 2 ⇒    α - 6 = 2 , - 2 ⇒    α = 8 , 4 d 2 = β 3 - 3 2 2 + 1 2 = 2 5    ⇒    | β - 9 | = 6 ⇒    β - 9 = 6 ,