Google Classroom
GeoGebraClasse GeoGebra

circle tangent to an ellipse and a line by one of its points

this work uses an implicit curve created with method of Lagrange multipliers https://en.wikipedia.org/wiki/Lagrange_multiplier the line is supposed xAxis and the point in origin. other positions is a simple problem of transformations.
try another objects and positions with the same method ( and share please)