Let the point P represent z = x + i y , x , y ∈ R in the Argand plane. Let the curves C 1 and C 2 be the loci of P satisfying the conditions ( i ) 2 z + i z - 2 is purely imaginary and ( ii ) Arg z + i z + 1 = π 2 , respectively. Then the point of intersection of the curves C 1 and C 2 , other than the origin, is

Step-by-step solution

It is given that z = x + i y ,   x ,   y ∈ ℝ And 2 z + i z - 2 = 2 ( x + i y ) + i ( x - 2 ) + i y Re 2 z + i z - 2 = 2 x + i ( 2 y + 1 ) ( x - 2 ) + i y × ( x - 2 ) - i y ( x - 2 ) - i y 0 = 2   x ( x - 2 ) + y ( 2   y + 1 ) = 0 0 = x 2 + y 2 - 2 x + y 2     . . . i And, z + i z - i = x + i ( y + 1 ) ( x + 1 ) + i y × ( x + 1 ) - i y ( x + 1 ) - i y Now, arg z + i z - i = tan - 1 ( x + 1 ) ( y + 1 ) - x y x ( x + 1 ) + y ( y + 1 ) = π 2 x 2 + y 2 x + x + y =