Let complex numbers α and

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.

If $z_0 = x_0 + iy_0$ satisfies the equation $2|z_0|^2 = r^2 + 2$ , then |α| is equal to

(2013 Adv.)

Solution:- Intersection of circles, the basic concept is to equations simultaneously and using properties of modulus of complex numbers.

We know that,

$|z|^2 = |z||\overline{z}|$

and $|z_1 - z_2| = (z_1-z_2)(\overline{z_1}-\overline{z_2})$

Here, $(x - x_0)^2 + (y-y_0)^2 = r^2$

and $(x - x_0)^2 + (y-y_0)^2 = 4r^2$ could be written as

$|z-z_0|^2 = r^2$ and $|z-z_0|^2 = 4r^2$

Since, α and $1/\overline{\alpha}$ lies on first and second respectively.

∴ $|\alpha-z_0|^2 = r^2$

⇒ $(\alpha - z_0)(\overline{\alpha} - \overline{z_0}) = r^0$

⇒ $|\alpha|^2 - z_0\overline{\alpha} - \overline{z_0}\alpha + |z_0|^2 = r^2$ . . . (i)

and $|1/\overline{\alpha} - z_0|^2 = 4r^2$

⇒ $(1/\overline{\alpha}-z_0)(1/\alpha - \overline{z_0}) =4r^2$

⇒ $1/\alpha^2 - \dfrac{z_0}{\alpha}-\dfrac{\overline{z_0}}{\overline{\alpha}}+z_0^2 = 4r^2$

Since, $1/\alpha^2 - \dfrac{z_0\overline{\alpha}}{|\alpha|^2}-\dfrac{\overline{z_0\alpha}}{|\alpha|^2}+z_0^2 = 4r^2$

⇒ $1-z_0\overline{\alpha} - \overline{z_0}\alpha + |\alpha|^2|z_0|^2 = 4r^2|\alpha|^2$ . . . (ii)

On substracting equation (i) and equation (ii), we get

$(|\alpha|^2 - z_0\overline{\alpha} - \overline{z_0}\alpha + |z_0|^2)-(1-z_0\overline{\alpha} - \overline{z_0}\alpha + |\alpha|^2|z_0|^2 ) = r^2- 4r^2|\alpha|^2$

⇒ $(|\alpha|^2 - 1) + |z_0|^2(1 - |\alpha|^2) = r^2(1 - 4|\alpha|^2)$

⇒ $(|\alpha|^2-1)(1 - |z_0|^2)=r^2(1-4|\alpha|^2)$

⇒ $(|\alpha|^2-1)(1 - \dfrac{r^2 + 2}{2}) = r^2(1-4|\alpha|^2)$

Given that, $z_0 = \dfrac{r^2 + 2}{2}$

⇒ $(|\alpha|^2-1)(-\dfrac{r^2}{2}) = r^2(1-4|\alpha|^2)$

⇒ $|\alpha|^2-1= -2+8|\alpha|^2$

⇒ $7|\alpha|^2 = 1$

⇒ $|\alpha| = 1/\sqrt{7}$

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 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