Problem 15 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$2 x^{2}+4 \sqrt{3} x y+6 y^{2}-6 x-3=0$$

Step-by-Step Solution

Answer

The given equation represents a parabola.

Step 1: Identify the General Form

Observe that the given equation is of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) where \(A = 2\), \(B = 4\sqrt{3}\), \(C = 6\), \(D = -6\), \(E = 0\), and \(F = -3\). Conic sections are generally represented in this form.

Step 2: Calculate the Discriminant

Calculate the discriminant \( \Delta = B^2 - 4AC \). Substitute the values: \( B^2 = (4\sqrt{3})^2 = 48 \) and \( 4AC = 4 \times 2 \times 6 = 48 \). Thus, \( \Delta = 48 - 48 = 0 \).

Step 3: Determine the Conic Section

Refer to the conic sections discriminant condition: \( \Delta = 0 \) indicates that the conic section is a **parabola**. Since the discriminant in this equation is zero, the equation represents a parabola.

Key Concepts

Parabola

A parabola is a unique type of conic section that appears when a plane cuts parallel to the side of a right circular cone. One of the key characteristics of a parabola is its U-shape, which can open upward, downward, left, or right.
In terms of algebra, if you come across a quadratic equation in two variables (like the one in our exercise), you may be dealing with a parabola. The equation of a parabola typically lacks the term where both variables are multiplied (i.e., the term with the coefficient \( Bxy \)). However, when this form is encountered, the behavior of the parabola is further influenced by the discriminant (as we'll see shortly).
Parabolas are areas of focus due to their reflective properties which are utilized in technology like satellite dishes and the design of automobile headlights.

• The vertex of the parabola is the peak or the bottom-most point of the curve.
• The axis of symmetry is a vertical or horizontal line that divides the parabola into two mirror images.
• The focus is a point from which distances are equally measured along the curve.

Discriminant

In the realm of algebra and geometry, the discriminant is a very informative tool that helps determine the nature of roots of an equation, especially for conics.
The discriminant for our type of equation, which includes cross products like \( Bxy \), is expressed as \( \Delta = B^2 - 4AC \). It guides us in recognizing which conic section an equation describes without graphing it.
Here's how it works for different conic sections:

• If \( \Delta > 0 \), the equation represents a hyperbola.
• If \( \Delta = 0 \), as in our problem, the conic section is a parabola.
• If \( \Delta < 0 \), the equation defines an ellipse or a circle (which is a special type of ellipse).

This property of the discriminant frees us from needing to alter forms of complex equations by straightforwardly revealing the conic type through simple arithmetic.

General Form of Conics

The general form of conic sections is expressed by the equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). This offers a unified way of presenting the equations for circles, ellipses, hyperbolas, and parabolas.
Each conic section corresponds to specific values or combinations of the coefficients \( A, B, \) and \( C \). Observing these coefficients can tell a lot about what kind of conic section you have.
For clarity:

• The value of \( B \) plays a role in identifying the rotation of the conic section. If \( B = 0 \), there's no rotation, simplifying identification.
• A circle, for instance, meets the condition \( A = C \) and \( B = 0 \).
• In a parabola, either \( A \) or \( C \) (but not both) equals zero in the simplified oriented form.
• A hyperbola, recognizable often when \( \Delta > 0 \), signifies the presence of two different quadratic terms \( x^2 \) and \( y^2 \) which have opposite signs.

Ultimately, understanding the general form helps transform complex visual problems into more interactive algebraic puzzles.