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=x y z prove that (2x)/(1-x^2)+(2y)/(1-y^2)+(2z)/(1-z^2)=(2x)/(1-x^2)(2y)/(1-y^2)(2z)/(1-z^2)dot

If x+y+z= pi/2 , prove that : cos(x-y-z) + cos(y-z-x) + cos (z-x-y)-4cosx cosy cosz=0

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

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 y is the mean proportional between x and z, prove that : (x^(2) - y^(2) + z^(2))/(x^(-2) - y^(-2) + z^(-2)) = y^(4) .

If z = x +iy satisfies |z+1|=1 then

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