Let w = z z ¯ + k 1 z + k 2 i z + λ ( 1 + i ) , k 1 , k 2 ∈ ℝ . . Let R e ( w ) = 0 be the circle C of radius 1 in the first quadrant touching the line y = 1 and the y - axis. If the curve I m ( w ) = 0 intersects C at A and B , then 30 ( A B ) 2 is equal to _ _ _ _ _ _ _ .

Step-by-step solution

Given, w = z z ¯ + k 1 z + k 2 i z + λ ( 1 + i ) , k 1 , k 2 ∈ ℝ . . And R e ( w ) = 0 , So, taking z = x + i y   &   z ¯ = x - i y We get, w = x 2 + y 2 + k 1 x + i y + k 2 i x + i y + λ ( 1 + i ) ⇒ Re w = 0 ⇒ x 2 + y 2 + k 1 x - k 2 y + λ = 0 And, Im ( w ) = 0 ⇒ k 1 y + k 2 x + λ = 0 Now given R e ( w ) = 0 circle is touching y = 1 and y - axis, with radius 1 , so we have equation of circle x 2 + y 2 - 2 x - 4 y + 4 = 0 So, on comparing wit