In general, $$x^2+y^2+2gx+2fy+c=0$$ represents a circle with centre at $C(-g,-f)$. Equations of the form $$ax^2+2hxy+by^2=0$$ represents a pair of straight lines passing through origin. But the equation $2x^2+2y^2+xy=0$ represents just a point $(0,0)$.
When I came to know that it was not a pair of straight line equation I tried this: $$2x^2+2y^2+xy=2(x+y)^2-3xy=0$$ Using AM GM inequality, $(x+y)^2\ge 4xy$ $$2(x+y)^2\ge 8xy$$ $$3xy\ge 8xy$$ Which is only possible if $x=y=0$.
But is there an easy way to quickly check if the equation $ax^2+2hxy+by^2=0$ represents a point circle? After all, I used the above method only after knowing it's not a pair of straight lines.