If z=x+iy where x,y are real ,prove that |x|+|y|less than equal to 2|z| - Maths - Complex Numbers and Quadratic Equations

Question

If z=x+iy where x,y are real ,prove that |x|+|y|less than equal to
2|z|

Rishabh Mittal · Accepted Answer

We know , Square of any real number is always greater than or equal to zero
So,x-y2≥0x2+y2-2 x y≥0x2+y2≥2 x yAdding x2+y2 on both sides 2x2+y2 ≥x2+y2+2 x y2z2≥x+y2Both the term are positive , so we can take the square root without disturbing inequality
2 z≥x + y⇒x + y≤2 zHence Proved

If z=x+iy where x,y are real ,prove that |x|+|y|less than equal to 2|z|.