Let a circle C in the complex plane pass through the points z₁ = 3 +i , z₂ = -1+i 3 and z₃ = 2i . If z is a point on C such that (z) = - 2 3 , and a line L₁ passes through z and z₁ . A second line L₂ passes through z₂ and is perpendicular to L₁ . The complex number representing the other point of intersection of L₂ with the circle C is

Step-by-step solution

Let the points be z₁ = ( 3 , 1) , z₂ = (-1, 3 ) , and z₃ = (0, 2) . The moduli of these points are |z₁| = 3+1 = 2 , |z₂| = 1+3 = 2 , and |z₃| = 2 . Thus, the circle C is centered at the origin with radius 2 , and its equation is x^2 + y^2 = 4 . The point z lies on C and has (z) = - 2 3 . So, z = 2 ( (- 2 3 ) + i (- 2 3 ) ) = -1 - i 3 . The line L₁ passes through z = (-1, - 3 ) and z₁ = ( 3 , 1) . Slope of L₁ is m₁ = 1 - (- 3 ) 3 - (-1) = 1+ 3 3 +1 = 1 . Since L₂ is perpendicular to L₁ , its slope is m₂ = -1 . Line