If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3i, 2+5i, 3i$ lie on a circle,then the centre of the circle is

If $\cos \alpha + 2 \cos \beta + 3 \cos \gamma = 0$,$\sin \alpha + 2 \sin \beta + 3 \sin \gamma = 0$ and $\alpha + \beta + \gamma = \pi$,then $\sin 3 \alpha + 8 \sin 3 \beta + 27 \sin 3 \gamma$ is equal to

Let $z_1$ and $z_2$ be two complex numbers such that $\frac{z_1}{z_2} + \frac{z_2}{z_1} = 1$. Then

Let $A = \{ z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1 \}$ and $B = \{ z \in \mathbb{C} : \arg(\frac{z-1}{z+1}) = \frac{2\pi}{3} \}$. Then $A \cap B$ is

Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i| \geq \sqrt{5}$. If the complex number $z_0$ is such that $\frac{1}{|z_0-1|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}: z \in S\right\}$,then the principal argument of $\frac{4-z_0-\bar{z}_0}{z_0-\bar{z}_0+2i}$ is

Let $z_1$ and $z_2$ be two complex numbers and roots of the equation $z^2 + az + b = 0$. If $O$ is the origin such that $OA = OB$ and $a^2 = \lambda b \cos^2 \frac{\alpha}{2}$,where $\alpha$ is the angle $\angle AOB$,then $\lambda$ is equal to: