TITLE: Point circle from pair of straight line equation
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: This is a special case of what is known as quadratic forms.
If you rewrite $ax^2+2hxy+by^2=0$ using matrices as
$$\left[\matrix{x \\ y}\right]^T \left[\matrix{a & h \\ h & b}\right] \left[\matrix{x \\ y}\right] = 0
$$
you can perform a change of variables to the inner matrix's eigenvectors to get something on the form
$$\left[\matrix{x' \\ y'}\right]^T \left[\matrix{d_1 & 0 \\ 0 & d_2}\right] \left[\matrix{x' \\ y'}\right] = 0
$$
which expands to $d_1{x'}^2 + d_2{y'}^2 = 0$
From this it is easy to see if it represents a single point, two intersecting lines or a single line just from the signs of $d_1$ and $d_2$ (or if one is zero).
If you instead had something non-zero on the right, you could instead get an ellipse, a hyperbola, or two parallell lines.
Quadratic forms also work in higher dimensions, but this answer is getting long as it is.
If you want, you can probably read more in a standard linear algebra textbook.