Number of solutions of the equation `z^3+[3(barz)^2]/|z|=0` where z is a complex number is

Variable complex number z satisfies the equation |z-1+2i|+|z+3-i|=10 . Prove that locus of complex number z is ellipse. Also, find the centre, foci and eccentricity of the ellipse.

Let z_(1)" and "z_(2) be two roots of the equation z^(2)+az+b=0 , z being complex number further, assume that the origin, z_(1)" and "z_(2) form an equilateral triangle, then

Solve the equation z^3 + 8i = 0 , where z in CC .

Find the radius and centre of the circle z bar(z) - (2 + 3i)z - (2 - 3i)bar(z) + 9 = 0 where z is a complex number

Given that the complex numbers which satisfy the equation | z z ^3|+| z z^3|=350 form a rectangle in the Argand plane with the length of its diagonal having an integral number of units, then (a)area of rectangle is 48 sq. units (b)if z_1, z_2, z_3, z_4 are vertices of rectangle, then z_1+z_2+z_3+z_4=0 (c)rectangle is symmetrical about the real axis (d) a r g(z_1-z_3)=pi/4or(3pi)/4

If complex number z(z!=2) satisfies the equation z^2=4z+|z|^2+(16)/(|z|^3) ,then the value of |z|^4 is______.

A complex number z satisfies the equation |Z^(2)-9|+|Z^(2)|=41 , then the true statements among the following are

If z=((1+isqrt3)^2)/(4i(1-isqrt3)) is a complex number then |z| is