Along the same line, I first binarize the image to obtain the ellipse, and rotate the image to get the correct origin of the pixel coordinates (we should recall that the pixel coordinates origin are at the left top corner):

imgbw = Binarize[ImageRotate[img, 270 Degree],Method -> {"BlackFraction", 0.99}];
elip = N@Position[ImageData@imgbw, 1];
Now, we fit the best ellipse by means of 51491:
lin = {#1^2, #1, #2, 2 #1 #2, #2^2} & @@@ elip;
lm = LinearModelFit[lin, {1, a, b, c, d}, {a, b, c, d}];
pa = lm["BestFitParameters"];
w[x_, y_] := pa.{1, x^2, x, y, 2 x y} - y^2;
ContourPlot[w[x, y] == 0,
{x, 200, 400}, {y, 200, 500}, ContourStyle -> {Black, Dashed},
Epilog -> {Red, , PointSize[0.003], Point[elip]}]

$w[x,y]=-3.06274 x^2-y^2+1192.22 x+152.71 y+1.829648 xy-196494=0$