Complex Numbers...

If $$z + \sqrt {2}|z + 1| + i = 0$$ and $$z = x + iy$$, then

If $$(a, 0)$$ is a point on a diameter segment of the circle $$x^2+y^2=4$$, then $$x^2-4x-a^2=0$$ has.

Let $$P(e^{i\theta_1})$$,  $$Q(e^{i\theta_2})$$  and  $$R(e^{i\theta_3})$$ be the vertices of a triangle PQR in the Argand Plane. The orthocenter of the triangle PQR is 

Let $$z,\omega$$ be complex numbers such that $$\vec{z}+i\vec{\omega}=0$$ and $$Arg(z\omega)=\pi$$ then $$Arg(z)=$$.

Study the statements carefully.
Statement I: Both the roots of the equation $$x^2-x+1=0$$ are real.
Statement II: The roots of the equation $$ax^2+bx+c=0$$ are real if and only if $$b^2-4ac \geq 0$$.
Which of the following options hold?

If '$$\omega$$' is a complex cube root of unity,then $$\omega ^{ \begin{pmatrix} \frac { 1 }{ 3 }  & +\frac { 2 }{ 9 } +\frac { 4 }{ 27 } ...\infty  \end{pmatrix} }+\omega^{ \begin{pmatrix} \frac { 1 }{ 2 }  & +\frac { 3 }{ 8 } +\frac { 9 }{ 32 } ...\infty  \end{pmatrix} }=$$

If $${z}_{1}=-3+5i;{z}_{2}=-5-3i$$ and $$z$$ is a complex number lying on the line segment joining $${z}_{1}$$ and $${z}_{2}$$, then $$arg(z)$$ can be:

If $$\dfrac {lz_{2}}{mz_{1}}$$ is purely imaginary number, then $$\left |\dfrac {\lambda z_{1} + \mu z_{2}}{\lambda z_{1} - \mu z_{2}}\right |$$ is equal to

The complex numbers $$z=x+iy$$ which satisfy the equation
$$\left| \cfrac { z-5i }{ z+5i }  \right| =1$$ lie on:

Find a complex number z satisfying the equation $$z+\sqrt{2}|z+1|+i=0$$