Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square. $$ 4 x^{2}-4 y^{2}-2 x+2 y+1=0 $$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Group and Rearrange Terms

Start by grouping the terms with similar variables together in the equation. The equation becomes:\[ 4x^2 - 2x - 4y^2 + 2y = -1 \] Next, we'll work on completing the square separately for the \(x\) and \(y\) terms.

Step 2: Factor and Complete the Square for x-Terms

For the \(x\) terms, factor out the coefficient of \(x^2\), which is 4:\[ 4(x^2 - \frac{1}{2}x) \]To complete the square, take half of the coefficient of \(x\) (which is \(-\frac{1}{2}\)), square it (\((\frac{-1}{4})^2 = \frac{1}{16}\)), and add and subtract \(\frac{1}{16}\) inside the parentheses:\[ 4 \left( x^2 - \frac{1}{2}x + \frac{1}{16} - \frac{1}{16} \right) \]Simplify to:\[ 4 \left( \left(x - \frac{1}{4}\right)^2 - \frac{1}{16} \right) \]

Step 3: Factor and Complete the Square for y-Terms

For the \(y\) terms, factor out the coefficient of \(y^2\), which is \(-4\):\[ -4(y^2 - \frac{1}{2}y) \]Complete the square by taking half of the coefficient of \(y\) (which is \( \frac{1}{2} \)), square it \((\frac{1}{4})^2 = \frac{1}{16} \), and add and subtract \(\frac{1}{16}\) inside the parentheses:\[ -4 \left( y^2 - \frac{1}{2}y + \frac{1}{16} - \frac{1}{16} \right) \]Simplify to:\[ -4 \left( \left(y - \frac{1}{4}\right)^2 - \frac{1}{16} \right) \]

Step 4: Combine and Simplify the Equation

Now, substitute the completed squares back into the equation:\[ 4 \left( \left(x - \frac{1}{4}\right)^2 - \frac{1}{16} \right) - 4 \left( \left(y - \frac{1}{4}\right)^2 - \frac{1}{16} \right) = -1 \]Distribute and simplify:\[ 4 \left(x - \frac{1}{4}\right)^2 - \frac{1}{4} - 4 \left(y - \frac{1}{4}\right)^2 + \frac{1}{4} = -1 \]Rearranging provides:\[ 4 \left(x - \frac{1}{4}\right)^2 - 4 \left(y - \frac{1}{4}\right)^2 = -1 \]

Step 5: Identify the Conic Form

Observe that the equation \[ 4 \left(x - \frac{1}{4}\right)^2 - 4 \left(y - \frac{1}{4}\right)^2 = -1 \] resembles the form \(A(u^2) - B(v^2) = C\), where \(A\), \(B\), and \(C\) are constants and \(A\) and \(B\) are positive.This is in the standard form of a hyperbola. Therefore, the conic section represented is a hyperbola.

Key Concepts

Conic Sections

Conic sections are geometric figures that are formed by the intersection of a plane and a cone. They are so called because of how they can be generated in this manner. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each has distinct characteristics and equations associated with them.

• A circle is formed when the plane cuts the cone parallel to its base. Its equation is of the form \( x^2 + y^2 = r^2 \) where \( r \) is the radius.
• An ellipse occurs when the intersection makes an oval shape and is not parallel to the base, represented by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• A parabola results when the plane is parallel to the side of the cone, yielding an equation such as \( y = ax^2 + bx + c \).
• A hyperbola is formed when the plane cuts through the cone at an angle such that both nappes are intersected, given by \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

Each type of conic section has unique properties regarding the foci, vertices, and eccentricity that differentiate them from one another. Recognizing these forms is crucial for solving equations related to geometry and algebra.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square trinomial. This is especially useful when dealing with equations of conic sections, as it can help simplify and rearrange terms into a recognizable form. Here’s how it typically works:

• Start with a quadratic in the form \( ax^2 + bx + c \).
• Reorganize the equation by grouping the \( x \) or \( y \) terms together.
• Factor out any coefficients of \( x^2 \) or \( y^2 \) if needed.
• Take half of the linear coefficient, square it, and then add and subtract this value within the equation.
• Rewrite the quadratic as a binomial squared that represents the completed square.

When applied to the original exercise, these steps helped transform the given equation into a standard form representing a conic section. Completing the square efficiently converts complex equations into forms that can be more easily recognized and analyzed.

Equation Manipulation

Equation manipulation involves the strategic use of algebraic techniques to alter and simplify equations. This is often necessary to understand the structure of equations or to transform them into a desired form, such as the standard form of a conic section. Key techniques for manipulating equations include:

• Rearranging terms to isolate variables or group similar terms together.
• Factoring to simplify terms or to complete the square.
• Applying algebraic operations like addition, subtraction, multiplication, or division.

For the original exercise, manipulating the equation by factoring and completing the square allowed us to rearrange it into the standard form of a hyperbola. This step-by-step manipulation provided clarity and eased the identification of the conic section. Mastering equation manipulation is a foundational skill in algebra and geometry that enables solving problems involving conic sections more efficiently.