Problem 16 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$-x^{2}+4 \sqrt{2} x y+2 y^{2}-2 y+1=0$$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Identify the General Form

The general form of a conic section is given by: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]For the equation \(-x^2 + 4\sqrt{2}xy + 2y^2 -2y + 1 = 0\), we can identify the coefficients: \(A = -1\), \(B = 4\sqrt{2}\), \(C = 2\), \(D = 0\), \(E = -2\), and \(F = 1\). Let’s note the values of \(A\), \(B\), and \(C\) as they will help in identifying the conic section.

Step 2: Compute the Discriminant

The discriminant, \(\Delta\), of a conic section is calculated using the formula:\[ \Delta = B^2 - 4AC \]Substituting the values we found: \[ \Delta = (4\sqrt{2})^2 - 4(-1)(2) = 16\cdot2 + 8 = 32 + 8 = 40 \]

Step 3: Determine the Conic Section

Based on the discriminant:- \(\Delta < 0\) : Ellipse- \(\Delta = 0\) : Parabola- \(\Delta > 0\) : HyperbolaSince \(\Delta = 40 > 0\), the given equation represents a hyperbola.

Key Concepts

Understanding the Hyperbola

A hyperbola is one of the key conic sections and is formed by the intersection of a double cone and a plane that cuts through both halves of the cone. Unlike ellipses and circles, hyperbolas consist of two separate curves known as branches. These branches are symmetrical about a central point called the center.

A hyperbola has distinct properties that make it unique:

• The two branches of a hyperbola open away from each other, either horizontally or vertically.
• It has two axes of symmetry: the transverse and conjugate axes.
• Hyperbolas have two focal points located inside each branch.
• The equation of a hyperbola can take multiple forms based on its orientation.

In our example, the significant positive value of the discriminant indicates the presence of a hyperbola. Thus, if you ever come across an equation and need to identify it, remembering these geometric traits will be incredibly helpful.

Exploring the Discriminant

The discriminant is a crucial piece of algebraic information that helps us differentiate between various conic sections. The discriminant for conic sections is determined by the formula:

• \( \Delta = B^2 - 4AC \)

In the context of conic sections, the discriminant serves as a "decision-maker":

• When \( \Delta < 0 \), the conic is an ellipse or a circle.
• When \( \Delta = 0 \), the conic represents a parabola.
• When \( \Delta > 0 \), the conic turns out to be a hyperbola.

In our exercise, substituting the values, \( B^2 \) for the coefficient of the \( xy \) term results in \( (4\sqrt{2})^2 \) and combining it with \( 4AC \) from the coefficients of \( x^2 \) and \( y^2 \) respectively, leads to determining \( \Delta = 40 \), denoting a hyperbola thanks to its positive result.

The General Form of Conic Sections

Conic sections such as circles, ellipses, parabolas, and hyperbolas can be represented in a unified algebraic form known as the general form. This expression encompasses all conic sections and is written as:

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

In this equation, \( A \), \( B \), and \( C \) are the coefficients that primarily influence the shape and nature of the conic.

The presence of the \( Bxy \) term is particularly telling, as

• its presence indicates a rotated conic section when combined with non-zero \( A \) and \( C \) values.

To identify which conic section you are dealing with, examine the coefficients:

• If \( A = C \) and \( B = 0 \), it's likely a circle.
• If \( A eq C \), a decision between ellipse or hyperbola can be further clarified by the discriminant.

So, recognizing the general formula and those key coefficients helps in directing us to the nature of the conic—just as in the original exercise where a hyperbola was identified.