{a Cos[t] Cos[Theta] - b Sin[t] Sin[Theta], a Cos[t] Sin[Theta] + b Sin[t] Cos[Theta]}

{u Cos[phi]+v Sin[phi]-xe,−u Sin[phi]+v Cos[phi]-ye}

Solve[k (u Cos[t] + v Sin[t] - xe) + d - ye == -u Sin[t] +
  v Cos[t], v][[1, 1]] /. {u -> x, v -> y, t -> -ArcTan[k]} // Simplify

yl=(d - k xe - ye)/Sqrt[1 + k^2]

phi=ArcTan[x2 - x1, y2 - y1]=ArcTan[k]

{a Cos[t] Cos[Theta-phi] - b Sin[t] Sin[Theta-phi],
 a Cos[t] Sin[Theta-phi] + b Sin[t] Cos[Theta-phi]}

Solve[a Cos[t] Sin[Theta-phi] + b Sin[t] Cos[Theta-phi] == yl, t] /. C[1] -> 0 // Simplify

Integrate[ Sqrt[(a Cos[t] Cos[Theta-phi] - b Sin[t] Sin[Theta-phi])^2 +
 (a Cos[t] Sin[Theta-phi] + b Sin[t] Cos[Theta-phi])^2] // Evaluate, {t, t1, t2},
 Assumptions -> {{a, b, t1, t2, Theta, phi} [Element] Reals, t2 > t1,a>0,b>0}]

a (-EllipticE[t1, 1 - b^2/a^2] + EllipticE[t2, 1 - b^2/a^2]) -  yl L /2

ellipse = TransformedRegion[
   Ellipsoid[{14, 14}, {5, 2}],
   RotationTransform[[Pi]/3]
   ];
{p1, p2} = {{-2, 17}, {-8, 16}};
Graphics[{Red, ellipse, Black, Line@{p1, p2}, Green, PointSize[Large],
   Point@{p1, p2}}, Frame -> True]

ArcLength@RegionIntersection[InfiniteLine[{p1, p2}], ellipse]
N@%
(* (40 Sqrt[259 (-6425 + 3828 Sqrt[3])])/(2881 - 252 Sqrt[3]) *)
(* 3.77312 *)

Graphics[{Red, ellipse, Black, Opacity[0.5],
  HalfPlane[{p1, p2}, {0, -1}]}, Frame -> True]

N@
 Area@RegionIntersection[HalfPlane[{p1, p2}, {0, -1}], ellipse]
(* 3.81495 *)