Clear["`*"]

eqns = {x^2/a^2 + y^2/b^2 == 1, y == k (x - x0) + y0};  

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

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

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

xp = Inactive[Plus] @@ MonomialList[xpoly, {x}];

eqn = {xp == 0}

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

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

second = {x1 + x2, x1 x2, y1 + y2, y1 y2, 
   y1 y2/(x1 x2), (x1 + x2)/2, (y1 + y2)/2} // 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*)

area = 1/2 Chordlength Sqrt[intercept^2]/Sqrt[slope^2 + 1] // 
  FullSimplify

Try TraditionalForm. It may or may not do what you want. The display of polynomials or other formulas in the exact way one wishes can be frustrating.

Clear["`*"]

eqns = {x^2/a^2 + y^2/b^2 == 1, y == k (x - x0) + y0};  

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

polys = Numerator[Together[polyex]];

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

Inactive[Plus] @@ Reverse[CoefficientList[xpoly, x]*{1, x, x^2}]

the result is:

-a^2 b^2 + (b^2 + a^2 k^2) x^2 + a^2 k^2 x0^2 - 2 a^2 k x0 y0 + 
 a^2 y0^2 + x (-2 a^2 k^2 x0 + 2 a^2 k y0)

Thank you very much. This result is what I want.

If we want to get further results like this, what should we do?

(b^2 + a^2 k^2) x^2 + (y0 - kx0) 2 a^2 k x + (-a^2 b^2 + 
   a^2 k^2 x0^2 - 2 a^2 k x0 y0 + a^2 y0^2)

That is to extract the common factor of x ^ 2, x and constant terms