ValueError: intersection of ellipse

[Jinchao-Yu]

Env: sympy-1.3 (mpmath-1.0.0), on Linux
Issue: The computing of intersection of 2 ellipses can sometimes raise issues. For example,
el1 = Ellipse(Point(0, 0), 2, 1)
el2 = Ellipse(Point(5, 0), 2, 1)
el3 = Ellipse(Point(3, 5), 4, 3)
el1.intersect(el2) # good, no intersection, but has Complex solutions in maths.
el1.intersect(el3) # ValueError: Imaginary coordinates are not permitted

[sylee957]

I think this issue is caused by equation solver giving some overcomplicated solution containing imaginary number like

[(1013/54 - 40*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))/27 - 10*(-4 + sqrt(1338/25 - 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3) + 11664/(25*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))) - 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)))/2 + sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))/2)**2/27 - 40*sqrt(1338/25 - 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3) + 11664/(25*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))) - 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)))/27, -4 + sqrt(1338/25 - 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3) + 11664/(25*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))) - 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)))/2 + sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))/2), (1013/54 + 40*sqrt(1338/25 - 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3) + 11664/(25*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))) - 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)))/27 - 10*(-4 + sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))/2 - sqrt(1338/25 - 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3) + 11664/(25*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))) - 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)))/2)**2/27 - 40*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))/27, -4 + sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))/2 - sqrt(1338/25 - 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3) + 11664/(25*sqrt(669/25 + 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)) + 2*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3))) - 1476909/(1250*(1757521827/125000 + 6561*sqrt(30812995)*I/12500)**(1/3)))/2)]

And the solver's assumption checker is not properly working for this specific expression.

[RishobGuhaPatra]

Hey, I would love to help with this issue! Please guide me to what needs to be done!