Can the same equation be classified as a circle and an ellipse?

Question

I don't know whether the following equation as a circle, a parabola, an ellipse, or a hyperbola.

3x² + 3y² – 3x – 6y – 7 = 0

Answer 1

It's a circle.
The equation of an ellipse is x^2/a^2+y^2/b^2=1. For a circle, a=b, so the equation becomes
x^2/a^2+y^2/a^2=1 , or x^2+y^2=a^2. Now a is the
radius in a circle, so x^2+y^2=r^2. So yes, the same equation can be classified as a circle and an ellipse,
IF a=b

It's true that a circle is a special kind of ellipse, but in
math work, an equation like this is called a circle, not
a "special ellipse".
Here's a few tips:
Whenever you have a polynomial with the General
Form Ax^2+By^2+Cx+Dy+E=0
Circle if A=B (as in your problem)
Ellipse A and B are both positive
Hyperbola if A and B have different signs
Parabola if either A or B (but not both)=0

To get from the General Form to the Standard Form
you do so by completing the square.
Circle: (x-h)^2+(y-k)^2=r^2. A circle doesn't have
fractions.
Ellipse: (x-h)^2/a^2 + (y-k)^2/b^2=1. An ellipse has
the SUM of the two variable terms.
Hyperbola:(x-h)^2/a^2 - (y-k)^2/b^2=1. This one opens to the left and right.
(y-k)^2/b^2 - (x-h)^2/a^2=1. This one opens upward
and downward. A hyperbola has the DIFFERENCE
of the two variable terms.
Parabola: y^2=4ax or x^2=4ay. Has only one variable
squared.

Answer 2

3x² + 3y² – 3x – 6y – 7 = 0

The General Equation for a Conic is
ax^2 + bxy + cy^2 + dx + ey + f = 0
in our case
a = 3
b = 0
c = 3
Find
b^2 -4ac = 0 -4(3)(3) = -36 <0, this is either an ellipse or a circle.
A circle is a special case of an ellipse.
3x² + 3y² – 3x – 6y – 7 = 0
3(x-1/2)^2 -3/4 +3(y -1)^2 -3 -7
(x-1/2)^2 +(y-1)^2 = 43/12 , this is a circle with center (1/2, 1) and radius =
sqrt(43/12)

Answer 3

A circle is a special kind of ellipse in which the major and minor axes are the same. So any circle is also an ellipse. In the same way any square is also a rectangle.

So the equation is both a circle and an ellipse.

Answer 4

This might be similar to the infamous squares and rectangles, but I don't know whether it is.

So, no. A circle is a circle. That's why they have a specific name for it. Whether or not it could technically ALSO and less specifically be called an ellipse is debatable, and not, as far as I'm aware, something you absolutely need to know.

In short:

A circle is a circle, not just an ellipse. You would never call it that.

Answer 5

Circle. The main point here is the x^2 and y^2 terms. As long as you see these, they're circles :)

Answer 6

no, this is a circle. It is a circle because x^2 and y^2 has the same coeffient.