Consider the conic section given by the equation $$400192 x^2 - 266112 xy + 177808 y^2 - 6212480 x + 4180640 y + 25085200 = 0$$ Then an appropriate rotation of coordinate axes to eliminate the $xy$ term is given by the equations
$x =$$X +$$Y$
$y =$$X +$$Y$
After applying this rotation, we obtain the following equation in $X$ and $Y$

(Do NOT simplify the equation you get by multiplying or dividing by any factor.)
Which conic section is it? (Acceptable answers are: ellipse, hyperbola and parabola.)
Answer:
The first focus of this conic has $xy$ coordinates (,). (Order the foci lexicographically according to the order of their $x$ and $y$ coordinates, ie. $(-1,5)$ precedes $(3,-2)$ and $(2,1)$ precedes $(2,4)$.)
The second focus of this conic has $xy$ coordinates (,). (If the conic is a parabola, just repeat the coordinates of the first focus.)
The equation of the directrix is $=0$. (If the conic is a parabola, write the directrix in the form $X-p=0$ or $Y-p=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$. If the conic is an ellipse or hyperbola, write the equation $1=0$, an impossible equation.)
The axis of the conic has equation $=0$. (The axis of a conic is the line joining the foci and the vertices. For an ellipse this is also known as the major axis. Write the equation in the form $X-c=0$ or $Y-c=0$, then apply an appropriate rotation of axes to get an equation in $x$ and $y$.)
The asymptote of smaller slope has equation $y=$ . (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.)
The asymptote of larger slope has equation $y=$ . (If the conic is not a hyperbola put $y+1$ on the right hand side of the equation, giving an impossible equation.)