Question

if z=x+iy and | 2z-1|=|z+1| , show that x square + y square = 2x​

Answer 1

Step-by-step explanation:

Given:

$z = x + iy \: \: and \: \: |2z - 1| = |z + 1|$

To Prove:

${x}^{2} + {y}^{2} = 2x$

Proof: Since, z = x + iy,

so, 2z - 1 = (2x - 1) + 2yi and z + 1 = (x + 1) + iy

Now,

$|2z - 1|^{2} = |z + 1|^{2}$

$= > |(2x - 1) + 2yi|^{2} = |(x + 1) + iy|^{2}$

$= > {(2x - 1)}^{2} + {(2y)}^{2} = {(x + 1)}^{2} + {(y)}^{2}$

$= > 4 {x}^{2} - 4x + 1 + 4 {y}^{2} = {x}^{2} + 2x + 1 + {y}^{2}$

$= > 3 {x}^{2} - 6x + 3 {y}^{2} = 0$

$= > {x}^{2} - 2x + {y}^{2} = 0$

$= > {x}^{2} + {y}^{2} = 2x$

Hence proved