# Intersection between rotated & translated ellipse and line

I have a rotated ellipse, not centered at the origin, defined by x,y,a,b and angle.
Then I have a segment defined by two points x1,y1 and x2,y2
Is there a quick way to find the intersection points?

I used wolfram alpha equation solver, I tried to insert the equation of a line into the one of a standard non rotated, non translated ellipse,

and resolving for x this is the result

which is nice.

Then I took the equation of a rotated and translated ellipse

and this is the result

Which is obviously impractical, can anyone suggest a different method?

Simon Bridge