For three consecutive odd integers a , b and c , if the variable line a x + b y + c = 0 always passes through the point ( α , β ) , the value of α 2 + β 2 equals

Step-by-step solution

The given line is a x + b y + c = 0 . Since a ,   b and c are three consecutive odd integers, so they are in arithmetic progression, ⇒ b - a = c - b ⇒ c = 2 b - a Hence, the equation of line is a x + b y + 2 b - a = 0 ⇒ a x - 1 + b y + 2 = 0 . Now, the is given that the line always pass through point α ,   β ⇒ a α - 1 + b β + 2 = 0 This is only possible when, α - 1 = 0 and β + 2 = 0 ⇒ α = 1 and β = - 2 ⇒ α 2 + β 2 = 1 2