If α denotes the number of solutions of 1 − i x = 2 x and β = z arg z , where z = π 4 1 + i 4 1 − π · i π + i + π − i 1 + π · i , i = − 1 , then the distance of the point α , β from the line 4 x − 3 y = 7 is ______

Step-by-step solution

Given, 1 − i x = 2 x ⇒ 2 x = 2 x ⇒ x = 0 Hence, the number of solutions will be one, so α = 1 Now, solving z = π 4 1 + i 4 1 − π · i π + i + π − i 1 + π · i ⇒ z = π 4 1 + i 4 π − π i − i − π π + 1 + π − i − π i − π 1 + π ⇒ z = π 4 1 + i 4 − 2 π i − 2 i π + 1 ⇒ z = π i 2 1 + i 4 ⇒ z = − π i 2 1 + 4 i + 6 i 2 + 4 i 3 + 1 ⇒ z = 2 π i So, a r g z = π 2 ⇒ β = z a r g z = 2 π π 2 = 4 Hence, the distance from α , β ≡ 1 , 4 to 4 x − 3 y = 7 will be 4 - 3 × 4 - 7 4 2 + 3 2 = 15 5 = 3