Question

What conics do the following equations represent? When possible, find their centres, and also their equations referred to the centre.
$y^2-2\sqrt{3}xy+3x^2+6x-4y+5=0.$

Answer 1

$h^2-ac=-2<0\Rightarrow$ Given equation can be for ellipse/point/No curve.

For the centre of ellipse $\frac{\delta S}{\delta x}=\frac{\delta S}{\delta y}=0$

$2y-2\sqrt{3}x-4=06x-2\sqrt{3}y+6=0$

No solution for x and y

this represents No curve.