If w = α + iβ where β ≠ 0 and z ≠ 1 satisfies that

Question 3:- If w = α + iβ where β ≠ 0 and z ≠ 1 satisfies that condition that $\left(\dfrac{w - \overline{w}z}{1 - z}\right)$ is purely real, then the set of values of z is

(a) |z| = 1, z ≠ 2 (b) |z| = 1 and z ≠ 1

(c) $z = \overline{z}$ (d) None of these [JEE 2006]

Solution:- Let $z_1=\left(\dfrac{w - \overline{w}z}{1 - z}\right)$ be purely real ⇒ $z_1 = \overline{z_1}$

∴ $\left(\dfrac{w - \overline{w}z}{1 - z}\right) = \left(\dfrac{\overline{w} - w\overline{z}}{1 - \overline{z}}\right)$

⇒ $w - w\overline{z} - \overline{w}z + \overline{w}z.\overline{z} = \overline{w} - z \overline{w}-w\overline{z}+wz.\overline{z}$

⇒ $(w-\overline{w}) + (\overline{w}-w)|z|^2=0$

⇒ $(w -\overline{w})(1 - |z|^2) = 0$

⇒ |z|² = 1 [as $w - \overline{w} \neq 0,$ since β ≠ 0]

⇒ |z| = 1 and z ≠ 1

Question 1:- If z is complex number such that |z| ≥ 2 . then the minimum value of |z + 1/2|.

(a) is equal to 5/2

(b) lies int interval (1, 2)

(c) is strictly greater than 5/2

(d) is strictly greater than 3/2 but less than 5/2

Solution:- See full solution

Question 2:- Let complex numbers α and $1/\overline{\alpha}$ lies on circles $(x - x_0)^2 + (y-y_0)^2 = r^2$ and $(x - x_0)^2 + (y-y_0)^2 = 4r^2$ , respectively.