Let z₁ be a complex number satisfying |z₁ - 3 - 4i| = 2 . Let z₂ = z₁ ( 1+i 3 2 ) . The maximum possible area of the triangle formed by the origin O , z₁ , and z₂ is

Step-by-step solution

The relation between z₂ and z₁ is given by: z₂ = z₁ ( 1+i 3 2 ) = z₁ e^ i /3 This implies that |z₂| = |z₁| and (z₂) - (z₁) = 3 . Since two sides of the triangle O z₁ z₂ are equal in length and the included angle is 60^ , the triangle is equilateral with side length |z₁| . The area of this equilateral triangle is: Area = 3 4 |z₁|^2 To maximize the area, we need to maximize |z₁| . The locus of z₁ is a circle centered at C(3, 4) with radius r = 2 . The distance from the origin to the center of the circle is: OC = 3^2