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Chapter 12: Problem 31

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ 6 x^{2}+10 x y+3 y^{2}-6 y=36 $$

Short Answer

Expert verified
The conic is a hyperbola, confirmed by the discriminant and graphing.

Step by step solution

01

Write the General Form of the Conic

The given equation is:\[6x^2 + 10xy + 3y^2 - 6y = 36\]In the general quadratic form of a conic, which is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the coefficients for this equation are \(A = 6\), \(B = 10\), \(C = 3\), \(D = 0\), \(E = -6\), and \(F = -36\).
02

Calculate the Discriminant

The discriminant for a conic equation is given by \(B^2 - 4AC\). Substitute the values of \(B\), \(A\), and \(C\):\[B^2 - 4AC = 10^2 - 4(6)(3)\]\[= 100 - 72 = 28\]
03

Analyze the Discriminant

Use the discriminant to determine the type of conic:- If the discriminant is greater than 0, the conic is a hyperbola.- If the discriminant equals 0, the conic is a parabola.- If the discriminant is less than 0, the conic is an ellipse or circle.Since \(28 > 0\), the conic is a hyperbola.
04

Confirm by Graphing the Conic

Graph the conic equation \(6x^2 + 10xy + 3y^2 - 6y = 36\) using a graphing device. Observe if the graph resembles the shape characteristic of a hyperbola, which has two distinct branches opening either opposite or adjacent to each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Discriminant in Conic Sections
The discriminant is a useful tool in the study of conic sections. It's a number calculated from the coefficients of a quadratic equation. In this context, the equation is given as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The formula for the discriminant, \( B^2 - 4AC \), helps in identifying the type of conic section represented by the equation.

Here's how you can interpret the discriminant:
  • If \( B^2 - 4AC > 0 \), the conic is a hyperbola.
  • If \( B^2 - 4AC = 0 \), the conic is a parabola.
  • If \( B^2 - 4AC < 0 \), the conic is an ellipse or could be a circle if it meets further conditions.
This method offers a quick mathematical means to categorize conic sections without graphing them. For the equation \( 6x^2 + 10xy + 3y^2 - 6y = 36 \), the discriminant calculated was \( 28 \), indicating a hyperbola.
Recognizing a Hyperbola
A hyperbola is one of the fascinating conic sections and is recognized by its unique shape. It consists of two separate curves, called branches, that open in opposite directions.

When identifying a hyperbola from a quadratic equation, one indicator is the value of the discriminant explained earlier. If \( B^2 - 4AC > 0 \), it confirms the presence of a hyperbola.
  • The branches can open horizontally or vertically, depending on the equation's structure.
  • In the standard form, a hyperbola often has the term \( xy \) present, which affects its rotation and orientation on a graph.
Understanding the visual characteristics of a hyperbola will help when confirming its presence through graphing.
Basics of Quadratic Equations in Conics
Quadratic equations are foundational to conic sections, encompassing different forms like circles, ellipses, parabolas, and hyperbolas. The general quadratic equation for a conic section is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). This equation can represent various shapes depending on the values of \( A \), \( B \), and \( C \).
  • When \( B = 0 \) and \( A eq C\), the equation represents either an ellipse or a hyperbola, depending on other conditions.
  • A perfect square in \( y \) or \( x \) may hint at a parabola.
  • If \( A = C \) and \( B = 0 \), the conic is a circle.
Knowing these characteristics assists in predicting the conic's type before proceeding to graph it. Quadratic equations, being the skeleton of conics, provide a mathematical explanation for the shapes we see in graphs.
Graphing Conics to Confirm Their Nature
Graphing conic sections provides a visual confirmation of their nature, as it allows you to see the actual shape. Once you suspect a conic type via the discriminant, graphing checks if the shape corresponds to your conclusion.

To graph the conic equation \( 6x^2 + 10xy + 3y^2 - 6y = 36 \), a graphing tool or graphing calculator can be used. Upon graphing, examine these features:
  • For a hyperbola, observe if there are two branches that mirror each other and open in opposite directions.
  • Ensure the asymptotic lines, which guide the direction of the hyperbola's branches, are clear.
  • If the graph does not display as expected, recheck the calculation inputs or settings.
Double-checking via graphing ensures the reliability of analytical predictions based on the equation properties and provides a fuller understanding of conic sections.

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Most popular questions from this chapter

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 8),\) asymptotes: \(y=\pm \frac{1}{2} x\)

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ x^{2}-2 x y+3 y^{2}=8 $$

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