Question

How do you classify the conic $- 3{x^2} - 3{y^2} + 6x + 4y + 1 = 0$?

Answer 1

Here we have to classify the given expression on conic. So that we have to use the general form of a conic and compared to the given expression. On doing some classification we get the required answer.

Complete step by step answer:
It is given that, the conic is $- 3{x^2} - 3{y^2} + 6x + 4y + 1 = 0$
We have to classify the conic.
To classify the conic, we have to follow a few rules. These are:
the general form of a conic is $A{x^2} + B{y^2} + Dx + Ey + F = 0$
A and B cannot both equal zero - this would be the equation of a line
if A = B, the conic is a circle
if A or B = 0, the conic is a parabola
if A is not equal to B and AB > 0, the conic is an ellipse
if AB < 0, the conic is a hyperbola
The given conic satisfies the first condition that us $A = B = - 3$
So, this conic is a circle.

Note: A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone.
It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic.