<s>[INST] Where is the mistake? Finding an equation for the ellipse with foci $(1,2)$, $(3,4)$, and sum of distance to the foci equals to $5$. 
Find an equation for the ellipse with foci $(1,2)$, $(3,4)$, and sum of distance to the foci equals to $5$.

We consider the foci in the coordinate system $XY$ such that $X=x-2$ and $Y=y-1-x$, the coordinates of the foci in this system are $(-1,0)$ and $(1,0)$, furthermore $2a=5$, the equation of the ellipse in $XY$ is
\begin{equation}
\left( \frac{X}{2.5}\right)^2 + \left( \frac{Y}{\sqrt{5.25}} \right)^2 = 1
\end{equation}
and this can be expressed in $xy$ as
$$\left( \frac{x-2}{2.5} \right) + \left( \frac{y-1-x}{\sqrt{5.25}}\right) = 1$$
I have made the graph of the last equation and it is not the case that foci are $(1,2)$ and $(3,4)$, so, can anyone help me to see the mistake please?
 [/INST] As shown in other answer, you can just form an equation for sum of distances to foci. Going by your method, the goal would be to write
$$\left( \frac{\text{Distance from Line perp to Axis, thro' center }}{\text{Semi-major axis}} \right)^2 + \left( \frac{\text{Distance from Axis}}{\text{Semi-minor axis}} \right)^2 = 1$$
which are proper substitutes for $X,Y$ in the standard form.

*

*Axis is the line passing through foci : $y=x+1$

*Center of ellipse is midpoint of foci : $(2,3)$

*Line perpendicular to axis and passing through center $(2,3)$ is $x+y=5$

*You can calculate semi-major axis $=a$ and semi-minor axis $=b$.

Then required equation of ellipse is
$$\frac{1}{a^2}\left( \frac{x+y-5}{\sqrt{2}} \right)^2 + \frac{1}{b^2}\left( \frac{x-y+1}{\sqrt{2}} \right)^2 = 1$$
This simplifies to $$84x^2 -32xy + 84y^2 + \cdots = 0$$
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