The circle described on the major axis of an ellipse as diameter is called an auxiliary circle of the ellipse.
For the ellipse x²/a² + y²/b² = 1
Auxiliary circle is x² + y² = a²
Parametric coordinates of ellipse
If ф is the angle made by the line joining centre of ellipse with a point (x,y) on the ellipse x = a cos ф and y = b sin ф are the parametric coordinates of the ellipse x²/a² + y²/b² = 1, a>b