Problem 29 - Algebra 2

Question

LIGHT For Exercises 29 and \(30,\) use the following information. A lamp standing near a wall throws an arc of light in the shape of a conic section. Suppose the edge of the light can be represented by the equation \(3 y^{2}-2 y-4 x^{2}+2 x-8=0\) . Identify the shape of the edge of the light.

Step-by-Step Solution

Answer

The conic section is a hyperbola.

Step 1: Write the Conic Section Equation

The equation provided is \(3y^2 - 2y - 4x^2 + 2x - 8 = 0\). The goal is to identify the conic section that this equation represents based on its structure.

Step 2: Rearrange and Group Terms

Start grouping terms involving \(x\) and \(y\) together: \[-4x^2 + 2x + 3y^2 - 2y = 8\].

Step 3: Identify the Conic Section Form

Notice that the equation involves both \(x^2\) and \(y^2\) with opposite signs, which indicates the conic section is a hyperbola.

Key Concepts

Hyperbola

A hyperbola is one of the major types of conic sections, which are the curves formed by the intersection of a plane and a double-napped cone. Unlike ellipses and circles, which have a single connected curve, or parabolas, which have one open-ended curve, hyperbolas are distinct because they have two separate branches.

Here are some key aspects of hyperbolas:

• Defining Feature: A hyperbola is defined by its geometric property that for any point on the hyperbola, the absolute difference of the distances to two fixed points called foci is constant.
• Standard Equation: In standard form, the equation of a hyperbola centered at \( (h, k) \) can be written as:
  \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or
  \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \).
• Axes and Asymptotes: Hyperbolas have two axes, the transverse axis and the conjugate axis, and they also have asymptotes which are lines that the branches of the hyperbola approach as they extend to infinity.

Understanding these features helps when identifying hyperbolas from their equations, as the presence of negative signs between squared terms is a giveaway.

Equation of a hyperbola

The equation of a hyperbola exhibits certain characteristics that make it distinctive among conic sections. In the equation we examined, \(3y^2 - 2y - 4x^2 + 2x - 8 = 0\), it’s essential to note the signs and arrangements of the \(x^2\) and \(y^2\) terms, which can help identify the type of hyperbola specified.

To identify a hyperbola from an equation:

• Squared Terms: Look for terms that involve squares, i.e., \(x^2\) and \(y^2\). In our equation both \(3y^2\) and \(-4x^2\) appear.
• Opposite Signs: The coefficients in front of these squared terms must have opposite signs. It is the negative sign between them, \(-4x^2\) and \(3y^2\), which confirms the presence of a hyperbola.

For this equation, you can also group the terms and quickly rearrange them to reflect the structure common to standard hyperbolic equations. These steps highlight how quickly determining the type of conic section is manageable through recognizing the signs and arrangement of the squared variables.

Conic section identification

Conic sections are fundamental shapes in mathematics and geometry, each defined by a unique set of features. They are the curves obtained by slicing various angles through a cone. Understanding how to identify each is crucial in many fields like physics, engineering, and astronomy.

When it comes to identifying the conic sections in an equation:

• Key Components: The general equation for a conic is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The relationship between \(A\) and \(C\) determines the type of conic:
• Circle: If \(A = C\) with no \(B\) term, it's a circle.
• Ellipse: If \(A\) and \(C\) have the same sign but \(A eq C\), it describes an ellipse.
• Hyperbola: If \(A\) and \(C\) have opposite signs, the conic is a hyperbola.
• Parabola: If either \(A\) or \(C\) is zero (but not both), it’s a parabola.

In the equation provided, the involvement of both \(3y^2\) and \(-4x^2\) with opposite signs signifies a hyperbola. This foundation in identifying conic sections helps in analysis and applications across different disciplines.