If `z=x+iy` prove that ` |\x|+|\y|lesqrt(2)|\z| `

If z=x+iy,x,y real , then |x|+|y|lek|z| where k is equal to :

If x+y+z=xzy , prove that : x(1-y^2) (1-z^2) + y(1-z^2) (1-x^2) + z (1-x^2) (1-y^2) = 4xyz .

If x + y + z = xyz , prove that x(1 -y^(2)) (1- z^(2))+ y(1- z^(2))(1- x^(2)) +z(1-x^(2)) (1- y^(2)) = 4xyz .

If z=x+iy then prove that the solution of equation iz+2i+1=0 is x=-2,y=1

If z=x+ iy,x,y real,then |x|+|y|<=k|z| where k is equal to :

If z = x + iy and |z+6| = |2z+3| , prove that x^2+y^2 =9.

If complex number z=x +iy satisfies the equation Re (z+1) = |z-1| , then prove that z lies on y^(2) = 4x .

If x,y,z gt 0 and x + y + z = 1, the prove that (2x)/(1 - x) + (2y)/(1 - y) + (2z)/(1 - z) ge 3 .

If z=x+iy such that the argument of (z-1)/(z+1) is always (pi)/(4). Prove that x^(2)+y^(2)-2y=1