Problem 15 - College Algebra with Corequisite Support

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

The given equation is \(2 x^2 + 4 \sqrt{3} xy + 6 y^2 - 6x - 3 = 0\). A conic section equation is typically in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, \(A = 2\), \(B = 4\sqrt{3}\), \(C = 6\).

Step 2: Use the Discriminant Method

The discriminant of the conic section is given by \(B^2 - 4AC\). Substitute the values: \((4\sqrt{3})^2 - 4(2)(6)\).

Step 3: Calculate the Discriminant

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

Step 4: Determine the Conic Type

If the discriminant \(B^2 - 4AC = 0\), the conic section is a **parabola** as this condition signifies a degenerate form of a conic, typically a pair of lines or a single line.

Key Concepts

Discriminant Method

The discriminant method is a mathematical tool used to identify the type of conic section an equation represents. Conic sections include circles, ellipses, parabolas, and hyperbolas. Each of these shapes has unique properties and is distinguishable through their algebraic expression form.

To use the discriminant method, the equation of the conic section must be in the general form: \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\] where \(A\), \(B\), and \(C\) are constants that help define the curve’s nature.

The discriminant for conic sections is determined using the expression:\[B^2 - 4AC\]

• If the discriminant equals zero (\(B^2 - 4AC = 0\)), the conic is a parabola or could be degenerate like a line or pair of lines.
• If the discriminant is greater than zero, the conic section is a hyperbola.
• If it’s less than zero, the conic section might be an ellipse, and if \(A = C\) and \(B = 0\), it’s a circle.

Understanding the discriminant method is essential when analyzing the structure of equations to help determine their graphical representations.

Equation Transformation

Equation transformation involves manipulating the standard form of a conic section equation to identify or highlight certain properties. Transforming an equation can often make it easier to determine the type of conic section it describes.

Sometimes, equations are given in a form that isn't immediately recognizable as a standard conic section. Through various algebraic techniques such as completing the square, reversing the formula, or aligning coefficients, we can translate these into recognizable forms. For example, starting with a general quadratic equation like:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\] we can rearrange and simplify it.

This rewriting can be crucial in solving or sketching conics as it simplifies calculations and clarifies relationships between variables.

• For instance, a perfectly squared term can expose vertex information.
• Shifting or rotating axes might assist in analyzing symmetry or orientation.

Mastering these transformations ensures you can diagnose the curve's shape and properties efficiently.

Parabolas

Parabolas are unique among conics due to their singular axis of symmetry and their reflective property, which makes them fascinating. When an equation of a conic section is checked using the discriminant method and deemed to have a discriminant of zero (\(B^2 - 4AC = 0\)), it confirms the presence of a parabola.

Parabolas can open upwards, downwards, or to the sides. Their standard forms are:

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

Parabolas have a vertex, which acts as the point of symmetry. They also feature a focus and a directrix. The distance from any point on the parabola to the focus is equal to the distance from the point to the directrix.

This reflective property is why parabolas are prominent in designing satellite dishes and headlight reflectors. Solving problems involving parabolas often requires identifying these properties and using them effectively to analyze their geometric structure.