<s>[INST] How to find an ellipse equation with non-symmetrical foci values? I have tried a lot but i do not seem to find an equation.
Question is the following:
Find an equation of the specified curve, that is, of the locus of points the sum of whose distances from the points $(2,3)$ and $(4,1)$ is $8$.
I first tried to translate and rotate the axis to see the ellipse like its center lies on the origin $(0,0)$. It seemed to be tough. I know I can use the general equation with center $(h,k)$ but then there will not be any rotation though the major axis is rotated.
 [/INST] I don't know.  For problems like this, I just use the distance formula and grind through the algebra.  (Then double-check my solution against Wolfram Alpha.)
$$\sqrt{(x-2)^2+(y-3)^2}=8-\sqrt{(x-4)^2+(y-1)^2}\\
(x-2)^2+(y-3)^2=64+(x-4)^2+(y-1)^2-16\sqrt{{(x-4)^2+(y-1)^2}}\\
(x^2-4x+4)+(y^2-6y+9)=64+(x^2-8x+16)+(y^2-2y+1)-16\sqrt{{(x^2-8x+16)+(y^2-2y+1)}}\\
4x-4y-68=-16\sqrt{x^2+y^2-8x-2y+17}\\
x-y-17=-4\sqrt{x^2+y^2-8x-2y+17}\\
(x-y-17)^2=16(x^2+y^2-8x-2y+17)\\
x^2+y^2+289-2xy-34x+34y=16x^2-128x+16y^2-32y+272\\
15x^2+2xy+15y^2-94x-66y-17=0
$$
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