You might try the Conic command.
To make the curve pick point B then C then the intersection of B and C. After that you can select the shape of the curve.
The result curve will always be tangent to lines B and C.
Is there a specific reason it needs to be a mathematically accurate ellipse?
If not, draw a degree 3 curve using End and Near osnaps.
Start the curve at end point B. Place the second point using near and about half way down towards the guide curve intersection.
Third point about halfway between the guide curve intersection and point C.
Fourth point at C.
Turn on the new curve’s control points and adjust the inner two points along the guide curves to suit.