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Chapter 8: Problem 2

What kind of curve has the Cartesian equation $$ 4 x^{2}+24 x y+11 y^{2}=5 ? $$ (See Ex. 5 at the end of \(88.2\).)

Short Answer

Expert verified
The curve is a hyperbola.

Step by step solution

01

Understand the Standard Form of a Conic Section

Conic sections can be represented in the form of a general quadratic equation: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Our given equation is \( 4x^2 + 24xy + 11y^2 = 5 \). We see that \( A = 4 \), \( B = 24 \), \( C = 11 \), and \( F = -5 \) with \( D = 0 \) and \( E = 0 \).
02

Discriminant of a Conic Section

To identify the type of conic section, calculate the discriminant \( \, B^2 - 4AC \, \). If \( B^2 - 4AC = 0 \), it is a **parabola**; if \( B^2 - 4AC < 0 \), it is an **ellipse**; if \( B^2 - 4AC > 0 \), it is a **hyperbola**.
03

Calculate the Discriminant

For our equation, \( B = 24 \), \( A = 4 \), and \( C = 11 \). The discriminant is calculated as:\[ B^2 - 4AC = 24^2 - 4 \times 4 \times 11 \]\[ = 576 - 176 \]\[ = 400 \]Since \( 400 > 0 \), the condition for a hyperbola is satisfied.

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Key Concepts

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

Hyperbola
A hyperbola is one of the four basic types of conic sections, along with circles, ellipses, and parabolas. These curves are formed by the intersection of a plane and a double cone.
When the plane cuts both nappes (the upper and lower cones), the curve obtained is a hyperbola.
Key Features of a Hyperbola:
  • Two separate symmetrical curves called "branches."
  • A center point from which the curves are mirrored reflections.
  • Asymptotes: imaginary lines that the branches approach but never touch. These lines help define the overall shape of the hyperbola.
  • Focus and transverse axis, which are important for determining the shape and orientation of the hyperbola.
The general form of a hyperbola鈥檚 equation in Cartesian coordinates is usually \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
This general form helps us understand properties like orientation and the direction in which the hyperbola opens.
Cartesian Equation
Cartesian equations are equations that define geometric figures using Cartesian coordinates, a system developed by Ren茅 Descartes.
These equations involved variables with respect to the x and y axes laid out on a grid. By adjusting the coefficients and constants in these equations, we can represent various conic sections, including circles, ellipses, parabolas, and hyperbolas.
The general quadratic form of these equations is given by: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), with specific conditions determining the conic section type.
In our exercise, \( 4x^2 + 24xy + 11y^2 = 5 \) simplifies into a form where we can analyze its structure using the discriminant.
Discriminant of a Conic Section
The discriminant is a mathematical expression used to determine the type of conic section given its Cartesian equation. For a general quadratic equation of the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant is calculated as \( B^2 - 4AC \).
Here's what the discriminant tells us:
  • If \( B^2 - 4AC = 0 \), the conic is a parabola.
  • If \( B^2 - 4AC < 0 \), the conic is an ellipse.
  • If \( B^2 - 4AC > 0 \), the conic is a hyperbola.
For our specific equation, \( 4x^2 + 24xy + 11y^2 = 5 \), the discriminant calculation results in a value of 400, which is greater than 0. This confirms that the equation represents a hyperbola. Understanding the discriminant is essential for quickly identifying the specific shape and nature of conic sections within geometric problems.

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