My needs are shown in the picture is.

**1. Calculate the equation after a straight line and an ellipse are simultaneous (eliminate x)

1. Calculating the discriminant of simultaneous equations

2. Calculate Veda of simultaneous equations (x1+x2, x1 x2, y1+y2, y1 y2, x1 x2+y1 y2, x1 y2+x2 y1)

3. Calculate the chord AB length where the ellipse intersects the line**

I wrote some tedious code myself, and I think it can be optimized.

It is better to use the symbols in the picture to represent the corresponding results.

Such as：

x1+x2=-((2 a^2 k m)/(b^2 + a^2 k^2))

and so on

How can we further optimize it? Thank you!

The code is as follows:

eqns = {x^2/a^2 + y^2/b^2 == 1, y == k x + m};

polyex = Apply[Subtract, eqns, {1}];

polys = Numerator[Together[Apply[Subtract, eqns, {1}]]];

xpoly = Collect[Resultant[polys[[1]], polys[[2]], y], x]

discx = Factor[Discriminant[xpoly, x]]   (*discriminant*)

frist = Solve[eqns, {x, y}] // FullSimplify;
{{x1, y1}, {x2, y2}} = {x, y} /. frist;

second = {x1 + x2, x1 x2, y1 + y2, y1 y2} // FullSimplify

thrid = {x1 x2 + y1 y2, x1 y2 + x2 y1} // FullSimplify

slope = -CoefficientList[polyex[[2]], x][[2]];    (*k*)

intercept = -CoefficientList[CoefficientList[polyex[[2]], y][[1]], 
    x][[1]] ;  (*m*)

Chordlength = 
 FullSimplify[
  Sqrt[1 + slope^2] Sqrt[(x1 + x2)^2 - 4 x1 x2]]    (*AbsAB*)