Question

Find the name of the conic represented by the equation $x^2+y^2-2xy+20x+10=0$.

Answer 1

$x^2+y^2-2xy+20x+10=0$

Comparing with $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$

$a=1$

$b=1$

$h=-1$

$g=10$

$f=0$

$c=10$

$\therefore \Delta =\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=\left|\begin{array}{ccc}1& -1& 10\\ -1& 1& 0\\ 10& 0& 10\end{array}\right|$

$\Delta =-100<0$

Also $a\ne c$ then ellipse.