Problem 308 - Calculus Volume 3

Question

Determine which of the conic sections is described. $$ x^{2}+2 \sqrt{3} x y+3 y^{2}-6=0 $$

Step-by-Step Solution

Answer

The conic section is a parabola.

Step 1: Identify the general form of the conic section equation

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

Step 2: Determine the type of conic section using the discriminant

The discriminant \( \,\Delta \, \) of the conic section equation is given by \( B^2 - 4AC \). Substituting the values, we have \( \Delta = (2\sqrt{3})^2 - 4 \cdot 1 \cdot 3 = 12 - 12 = 0 \).

Step 3: Interpret the discriminant result

When the discriminant \( \,\Delta \, = 0 \), it indicates that the conic is a parabola. A quadratic equation with a zero discriminant corresponds to a parabola in the context of conic sections.

Key Concepts

Discriminant of conic sections

Conic sections, integral parts of algebra and geometry, can be determined by their discriminants. A discriminant is a value calculated from the coefficients of the conic's equation in general form. It plays a crucial role in identifying the specific type of conic section.

The general form of a conic section equation is written as:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

To find the discriminant, use the formula:

• \( \,\Delta \, = B^2 - 4AC \)

The discriminant's value tells us:

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

So, the discriminant acts as a categorical tool, guiding us in recognizing the underlying geometry described by the equation.

Parabolas

Among conic sections, parabolas are unique. They appear when the discriminant \( \,\Delta \, = 0 \). This condition implies a special relationship between the terms in the equation, making it distinct from hyperbolas and ellipses.

A parabola can be thought of as the set of all points equidistant from a fixed point, called the `focus`, and a line, called the `directrix`. This geometric property distinguishes it from other conic sections. In its basic form, a parabola graphs as a U-shaped curve opening either vertically or horizontally, depending on its equation. The standard forms are:

• Vertical: \( y = ax^2 + bx + c \)
• Horizontal: \( x = ay^2 + by + c \)

Understanding parabolas extends beyond mere plotting. They are crucial in fields such as physics and engineering, where parabolic mirrors and trajectories are common. Their simplicity, defined by the direct relationship of their axis of symmetry, makes parabolas fascinating and widely applicable.

General form of conic sections

The general form of conic sections is versatile, embracing a variety of geometric shapes under a single umbrella. No matter the form a conic takes, be it a circle, ellipse, parabola, or hyperbola, it can be expressed as:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

Analysing this form involves identifying its coefficients: \( A \), \( B \), \( C \), \( D \), \( E \), and \( F \). From there, the relationship these coefficients hold determines the conic type. Here is a quick breakdown:

• If \( B = 0 \) and \( A = C \), the equation represents a circle.
• If \( \,\Delta \, = B^2 - 4AC < 0 \) with \( A eq C \) or \( B eq 0 \), it forms an ellipse.
• If \( \,\Delta \, = 0 \), it's a parabola.
• If \( \,\Delta \, > 0 \), it's a hyperbola.

Each conic section, described by this general form, plays a unique role in mathematics, showcasing the diverse ways we can see and interpret equations representing shape and space.