# The equation $\left | z-i \right |=\left | z-1 \right |,i=\sqrt{-1},$ represents : Option 1)a circle of radius $\frac{1}{2}$.  Option 2)a circle of radius $1.$Option 3)the line through the origin with slope $1.$Option 4)the line through the origin with slope $-1.$

$\left | z-i \right |=\left | z-1 \right |$

$\left | x+iy-i \right |=\left | x+iy-1 \right |$

$\left | x+i\left ( y-1 \right ) \right |=\left | \left ( x-1 \right ) +iy\right |$

$\sqrt{x^{2}+\left ( y-1 \right )^{2}}= \sqrt{\left ( x-1 \right )^{2}+y^{2}}$

$x^{2}+\left ( y-1 \right )^{2}=\left ( x-1 \right )^{2}+y^{2}$

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

$-2y=-2x$

$y=x$

The line through the origin with slope 1

Option 1)

a circle of radius $\frac{1}{2}$.

Option 2)

a circle of radius $1.$

Option 3)

the line through the origin with slope $1.$

Option 4)

the line through the origin with slope $-1.$

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