Question

For the following equations, determine which of the conic sections is described. $$ x^{2}+2 \sqrt{3} x y+3 y^{2}-6=0 $$

Short answer

The equation describes a parabola.

Step-by-step solution

Identify the general form of the conic equation

The general form of a conic section is given by the equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Compare this with the given equation \( x^2 + 2 \sqrt{3} xy + 3y^2 - 6 = 0 \). Identifying coefficients: \( A = 1, B = 2\sqrt{3}, C = 3, D = 0, E = 0, F = -6 \).

Calculate the discriminant of the conic equation

The discriminant, \( \Delta \), helps determine the type of conic section. It is calculated using \( \Delta = B^2 - 4AC \). Substituting the identified coefficients, we get \( \Delta = (2\sqrt{3})^2 - 4 \cdot 1 \cdot 3 = 12 - 12 = 0 \).

Determine the conic section based on the discriminant

When the discriminant \( \Delta = 0 \), the conic section is a parabola. Since we've determined that \( \Delta \) for our equation is 0, the given equation represents a parabola.

Key concepts

Discriminant

To determine the type of conic section described by a given equation, we can use a special value called the discriminant. The discriminant for conic sections is defined as \(\Delta = B^2 - 4AC\), where \(A, B,\) and \(C\) are coefficients in the general form of a conic equation. The value of the discriminant gives us crucial information:

• If \(\Delta > 0\), the conic is a hyperbola.
• If \(\Delta = 0\), the conic is a parabola.
• If \(\Delta < 0\), the conic is an ellipse or a circle (specifically, a circle if \(A = C\)).

In the exercise, the given conic equation \(x^2 + 2 \sqrt{3} xy + 3y^2 - 6 = 0\) has coefficients \(A = 1, B = 2\sqrt{3}, C = 3\). Calculating the discriminant using these values, we find that \(\Delta = 0\), indicating that the conic section is a parabola.

Parabola

A parabola is a type of conic section that can be identified when the discriminant \(\Delta = 0\). It has a distinctive shape similar to a U or an upside-down U, depending on the vertex and orientation. Parabolas appear in many real-life situations, such as the path of a projectile or the shape of satellite dishes. In a conic section context:

• Parabolas have a vertex, which is their highest or lowest point.
• They consist of a symmetrical curve that opens in either the vertical or horizontal direction.
• The focus and directrix are two key elements that help define the parabola's specific shape and position.

In the given exercise, since \(\Delta\) equals zero, the conic section described by the equation \(x^2 + 2 \sqrt{3} xy + 3y^2 - 6 = 0\) is a parabola.

General Form of Conic Equation

The general form of a conic equation encompasses different types of conic sections, namely circles, ellipses, parabolas, and hyperbolas. This form is expressed as:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]Here:

• \(A, B, C, D, E,\) and \(F\) are constants or coefficients.
• Each conic section is characterized by specific values of these coefficients and their relationships.
• The term \(Bxy\) indicates the presence of rotation. When \(B = 0\), this means that the conic section has not been rotated from its standard orientation.

In the exercise, the provided equation \(x^2 + 2 \sqrt{3} xy + 3y^2 - 6 = 0\) can be matched with the general form by identifying:

• \(A = 1\)
• \(B = 2\sqrt{3}\)
• \(C = 3\)
• \(D = 0\)
• \(E = 0\)
• \(F = -6\)

These coefficients allow us to calculate the discriminant and determine the nature of the conic section, which in this case, is a parabola.