eqns = {x^2/a^2 + y^2/b^2 == 1, x == t y + n};

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

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

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

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

(*CoefficientList[polyex[[2]],y]*)

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

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

coeffs = CoefficientList[xpoly, y]

(*{rootproduct,rootsum}=Together[Most[coeffs]/Last[coeffs]]*)

{rootsum, 
  rootproduct} = {-CoefficientList[xpoly, y][[2]]/Last[coeffs], 
  CoefficientList[xpoly, y][[1]]/Last[coeffs]}   (*x1+x2;x1x2*)

{yrootsum, yrootproduct} = 
 Simplify[Together[{slope*rootsum + 2 intercept, 
    slope^2*rootproduct + intercept*slope*rootsum + 
     intercept^2}]] (*  y1+y2;y1y2*)

Simplify[rootproduct + yrootproduct]   (*x1x2+y1y2*)

Simplify[2 slope rootproduct + intercept rootsum]    (*x1y2+x2y1*)

Chordlength = 
 Simplify[Sqrt[1 + slope^2] Sqrt[
    rootsum^2 - 4 rootproduct]]    (*AbsAB*)

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 = Resultant[polys[[1]], polys[[2]], y]

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

(*CoefficientList[polyex[[2]],y]*)

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

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

coeffs = CoefficientList[xpoly, x]

(*{rootproduct,rootsum}=Together[Most[coeffs]/Last[coeffs]]*)

{rootsum, 
  rootproduct} = {-CoefficientList[xpoly, x][[2]]/Last[coeffs], 
  CoefficientList[xpoly, x][[1]]/Last[coeffs]}   (*x1+x2;x1x2*)

{yrootsum, yrootproduct} = 
 Simplify[Together[{slope*rootsum + 2 intercept, 
    slope^2*rootproduct + intercept*slope*rootsum + 
     intercept^2}]] (*  y1+y2;y1y2*)

Simplify[rootproduct + yrootproduct]   (*x1x2+y1y2*)

Simplify[2 slope rootproduct + intercept rootsum]    (*x1y2+x2y1*)

Chordlength = 
 Simplify[Sqrt[1 + slope^2] Sqrt[
    rootsum^2 - 4 rootproduct]]    (*AbsAB*)

eqns = {x^2/4 + y^2/2 == 1, y == (1/2) x + 1};

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

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

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

discx = Factor[Discriminant[xpoly, x]]

slope = -CoefficientList[polyex[[2]], x][[2]]

coeffs = CoefficientList[xpoly, x]

(*{rootproduct,rootsum}=Together[Most[coeffs]/Last[coeffs]]*)

{rootsum, 
  rootproduct} = {-CoefficientList[xpoly, x][[2]]/Last[coeffs], 
  CoefficientList[xpoly, x][[1]]/Last[coeffs]}

Chordlength = 
 Simplify[Sqrt[1 + slope^2] Sqrt[rootsum^2 - 4 rootproduct]]

slope = Coefficient[CoefficientList[polys[[2]], y], x, 1]

The problem is how to extract the linear equation y==kx+m that was originally input.

Coefficient k before x

the code above output the array

Thank you! But I don't understand the meaning of the answer obtained by this operation, which is different from the answer I want to get.

Thank you very much!

The result is equivalent to what I want. Can the result be more optimized?

The simultaneous equation requires the coefficient of quadratic term, the coefficient of primary term and the constant term are integral, not fractional. On this basis, the discriminant and Veda are calculated.

for example:

coeffs = CoefficientList[xpoly, x]

(* Out[579]= {-1 - m^2/b^2, -((2 k m)/b^2), 1/a^2 - k^2/b^2} *)

I want the result as follow:

a^2 b^2 + a^2 m^2 , 2 a^2 k m ,(-b^2 + a^2 k^2)  

the result of discriminant as well

Thank you again!

Thanks very much!

eqns = {x^2/a^2 + y^2/b^2 == 1, y == k x + m};
(*polys=Apply[Subtract,eqns,{1}]*)
polys = Numerator[Together[Apply[Subtract, eqns, {1}]]]

(*Out[575]={-1+x^2/a^2-y^2/b^2,-m-k x+y}*)

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

(*Out[578]=-((4 (-b^2+a^2 k^2-m^2))/(a^2 b^2))*)

coeffs = CoefficientList[xpoly, x]

(*Out[579]={-1-m^2/b^2,-((2 k m)/b^2),1/a^2-k^2/b^2}*)

{rootproduct, rootsum} = Together[Most[coeffs]/Last[coeffs]]

(*Out[581]={(a^2*(b^2+m^2))/(-b^2+a^2*k^2),-((2*a^2*k*m)/(b^2-a^2*k^2)\
)}*)
Chordlength = Simplify[Sqrt[rootsum^2 - 4 rootproduct]]

Veda still has an error in the calculation result, the result of the product of the two roots is correct, and the result of the sum of the two roots is wrong, because the sum of the two roots is equal to the opposite number of the coefficient of the first term than the coefficient of the second term, and the running result is one minus sign less.

I made this by myself. I feel too tedious and troublesome. How to optimize the results so that they can be output sequentially at once.