The ellipse is the stretch of a circle.

First of all inverse stretch the given ellipse into the circle.

This inverse stretch transforms point P into P'.

The searched ellipse is the direct transform of an ellipse containing the circle and P'.

This ellipse containing the circle and P' needs to have a curvature radius in A ≥ circle radius.

As the major axis is AP' = 2a, the curvature radius at vertex A being b²/a,

we get b = O'B from b² = R.a by a classical construction.

This gives the cyan ellipse.

Then the direct stretch transforms this ellipse into the searched one in red.