Using the Preservation of the Inverse Points Property to Find the Image of a Generalized Circle Under an Extended Mobius Transformation

Extended mobius transformations preserve the inverse points property, so that ifandare inverse points with respect to a generalized circlethenandare inverse points with respect to the image of the circle,under the mobius transformation.

We can use this inverse points property to find the equation of the image of a circle with a given equation under an extended mobius transformation.

Example: Find the equation of the image of the circlewith equation under the extended mobius transformation

The circlehas centreso thatand infinity are inverse points with respect to

Thenandare inverse points with respect to

The equation of the image ofmust then befor some

lies onso thatlies onandsatisfies

The equation ofis

Example: Find the equation of the image of the circlewith equation under the extended mobius transformation

The circlehas centreso thatand infinity are inverse points with respect to

Thenandare inverse points with respect to

The equation of the image ofmust then be

lies onso thatlies onandsatisfies

The equation ofis