Find an Linear Fractional Transforms that maps the upper half of a circle centered at (0,0) with radius 1 to the first quadrant.

Question

Now I know LFT’s map circles and lines to circles and lines.

This is asking for a mapping that maps an open set to another open set. My intuition tells me I should find 3 points on the circle that map to either the vertical or horizontals axis.

Then I can define the origin to be mapped to the a point inside the first quadrant and hopefully through some algebra, this would give me an LFT that maps as such.