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.

Answers (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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