Understand and use location of complex conjugates in the complex plane.

The Argand diagram below shows five labelled points. Given \(z = 3 + 2i\) and \(w = -1 + 4i\), which point represents \(\overline{z + w}\)?

Let z₁ = 3 + 4i and z₂ = -2 + 5i. A region R in the complex plane is defined by the set of all complex numbers w such that w lies inside or on the circle with diameter joining z̄₁ and z₂. (a) Determine the centre and radius of the circle that forms the boundary of region R. (2 marks) (b) Determine whether the point z̄₂ lies inside, on, or outside region R. (2 marks) (c) Find the value of (z₁ + z̄₁)/2 and determine whether this point lies in region R. (1 mark)

Let z = 3 + 4i and w = -2 + 5i. In which quadrant of the complex plane is the complex number z̄ + w̄ located?