Problem 44 - Algebra 2

Question

What type of conic section is represented by the equation \(y^{2}-6 y=x^{2}-8 ?\)

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Rearrange the Equation

Start by bringing all the terms to one side of the equation. The given equation is: \[ y^2 - 6y = x^2 - 8 \]Rearrange it to have all terms on one side:\[ y^2 - 6y - x^2 + 8 = 0 \]

Step 2: Identify the Terms

Identify the squared terms in the equation. We have:- \( y^2 \) as a squared term.- \( x^2 \) as a squared term.Both variables are squared.

Step 3: Analyze the Coefficients of the Squared Terms

In the rearranged equation \( y^2 - 6y - x^2 + 8 = 0 \), check the coefficients of the squared terms.- The coefficient of \( y^2 \) is \( 1 \).- The coefficient of \( x^2 \) is \( -1 \).The coefficients are different in sign.

Step 4: Determine the Type of Conic Section

When the square terms \( x^2 \) and \( y^2 \) have opposite coefficients, the conic section is a hyperbola. Therefore, the given equation represents a hyperbola.

Key Concepts

Hyperbola

A hyperbola is one of the central figures in the study of conic sections. It consists of two separate curves that open in opposite directions and possesses a distinctive set of properties. The important aspect of a hyperbola is that it describes the difference in distances to two focal points being constant.

• A hyperbola can be recognized when the coefficients of the squared terms have opposite signs in an equation (e.g., one is positive and the other negative).
• Just like an ellipse, a hyperbola has foci, which are essential in drawing its shape.
• The center of a hyperbola gives us the point equidistant from all the vertex points on its associated asymptotes.

In our case with the equation, the presence of opposite signs in the squares of the variables confirms that we're dealing with a hyperbola.

Equation Rearrangement

Rearranging an equation is about changing the structure to make it easier to interpret. In solving conic sections, placing all terms on one side allows us to clearly see the relationships between terms.
For the equation given:\[ y^2 - 6y = x^2 - 8\]We moved all terms to the left side to have:\[ y^2 - 6y - x^2 + 8 = 0\]This format helps in easily identifying the squared terms and their coefficients. Rearrangement often helps to identify the type of conic section by highlighting equation symmetry or differences between terms. The ultimate goal is usually to have a standard form equation that can be easily compared to known forms of conic sections.

Coefficients of Squared Terms

The coefficients of the squared terms in a conic section equation determine a lot about its type and position.

• In an equation with two squared terms, such as: \[y^2 - 6y - x^2 + 8 = 0\]the coefficients for these terms are key. Here, we have:
  • Coefficient of \( y^2 \): 1
  • Coefficient of \( x^2 \): -1
• When these coefficients differ in sign (one positive, one negative), the equation aligns with the structure of a hyperbola.
• Adjusting the coefficients can also lead to different types of conic sections, like an ellipse or a parabola.

Understanding the role of these coefficients aids in grasping why certain equations fit specific types of conic sections, like our identified hyperbola.

Identify Conic Sections

Identifying conic sections involves recognizing their unique properties and standard forms from equations. Conic sections can be circles, ellipses, parabolas, or hyperbolas. Each has distinct traits based on the arrangement and coefficients of squared terms.

• For a given equation, identify any squared terms such as \(x^2\) or \(y^2\).
• Look at the coefficients:
  • Same signs and equal coefficients generally indicate a circle.
  • Same signs but unequal coefficients typically suggest an ellipse.
  • Different signs, as seen in our exercise, point to a hyperbola.
  • If there's only one squared term, it might represent a parabola.

By examining these details, one can reliably classify and understand the type of conic section represented by an equation, enhancing both comprehension and solving skillset in algebra.