
$4x^2+9y^236=0$
An ellipse is one type of quadratic plane curve, or conic section. 

$13x^2+10xy+13y^272=0$
Ellipses can be oriented in any direction. They do not have to be
symmetric with either axis. 

$x^2+y^29=0$
A circle is also a quadratic plane curve, or conic section. A
circle can also be viewed as a very special ellipse. 

$x^2+y^22x+2y+2=0$
When the radius of a circle shrinks to zero, a degenerate circle will
result. In other words, a single point is one type of quadratic
plane curve. Interestingly, this graph can also be described as the union of two intersecting lines having slopes $\pm i$, each passing through the point $(1,1)$. 

$x^2+y^2+9=0$
Quadratic equations in two variables may have no real solutions.
Examining the equation, you might be inclined to describe this as a
circle with an imaginary radius. 

$x^2y^24=0$
A hyperbola is also a quadratic plane curve, or conic section. 

$xyxy=0$
Hyperbolas can be functions, if one of the asymptotes is vertical. 

$x^2xy2y^2+x+10y12=0$
Here is a degenerate hyperbola, which will result when the constant
term of a hyperbola is changed to zero. 

$x^2y2=0$
A parabola is also a quadratic plane curve, or conic section. 

$x^24xy+4y^216=0$
Quadratic equations in two variables can also produce the degenerate
case of two parallel lines. 

$x^2+2xy+y^26x6y+9=0$
Another degenerate case of a quadratic equation will produce a single
line, though with multiplicity two. This occurs when the equation
is a perfect square on one side, and zero on the other. 