Find the equation of the inverse of the unit circle

The equation of the unit circle is:

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$ x^2 + y^2 = 1 $

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To find the inverse, we use the transformation:

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$ z = \x0crac{1}{x + yi} $

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where $ z = u + vi $ and $ x + yi = \x0crac{1}{u – vi} $.

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Therefore, the inverse relation in terms of $u$ and $v$ becomes:

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$ u = \x0crac{x}{x^2 + y^2} = x $

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$ v = \x0crac{-y}{x^2 + y^2} = -y $

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Thus, the equation of the inverse of the unit circle is:

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$ u^2 + v^2 = 1 $

The unit circle equation is:

$ x^2 + y^2 = 1 $

Using the transformation:

$ z = x0crac{1}{x + yi} $

$ x + yi = x0crac{1}{u – vi} $

The inverse relationship becomes:

$ u = x $

$ v = -y $

The equation of the inverse unit circle is:

$ u^2 + v^2 = 1 $

The unit circle is:

$ x^2 + y^2 = 1 $

Using the transformation:

$ z = x0crac{1}{x + yi} $

The inverse relations are:

$ u = x $

$ v = -y $

The inverse unit circle equation is:

$ u^2 + v^2 = 1 $

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