## 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 if and are inverse points with respect to a generalized circle then and are 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 circle with equation under the extended mobius transformation The circle has centre so that and infinity are inverse points with respect to Then and are inverse points with respect to The equation of the image of must then be for some  lies on so that lies on and satisfies The equation of is Example: Find the equation of the image of the circle with equation under the extended mobius transformation The circle has centre so that and infinity are inverse points with respect to Then and are inverse points with respect to The equation of the image of must then be  lies on so that lies on and satisfies The equation of is  