Question

Write the equation of the hyperbola in standard form. \((4 y-x)(4 y+x)=4\)

Short answer

The standard form is \(\frac{y^2}{\frac{1}{4}} - \frac{x^2}{4} = 1\).

Step-by-step solution

Expand the given expression

Begin by expanding the expression \((4y - x)(4y + x)\). This is a difference of squares, which can be expanded as follows:\[(4y - x)(4y + x) = (4y)^2 - x^2 = 16y^2 - x^2\]So, the given equation becomes:\[16y^2 - x^2 = 4\]

Adjust the equation to match the standard form

In standard form, the equation of a hyperbola is usually written as:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad \text{or} \quad \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\]Transform the equation \(16y^2 - x^2 = 4\) into this form by dividing both sides by 4:\[\frac{16y^2}{4} - \frac{x^2}{4} = 1 \to 4y^2 - \frac{x^2}{4} = 1\]

Simplify the equation

Rewrite the equation in the proper standard form:\[\frac{y^2}{\frac{1}{4}} - \frac{x^2}{4} = 1\]Thus, this matches the standard form \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), where \(b^2 = \frac{1}{4}\) and \(a^2 = 4\). This is the standard form of the hyperbola equation.

Key concepts

Equation of Hyperbola

A hyperbola is a type of conic section that consists of two symmetrical open curves. The equation of a hyperbola can appear in various forms depending on its orientation and the given problem. In general, the difference of squares formula comes into play when deriving these equations. For example, if you have a product

• such as
• \((4y - x)(4y + x)\), you can expand it to find an equation as was done in the original problem solution.

Using the formula for the difference of squares, \((a - b)(a + b) = a^2 - b^2\), the expression expands to \(16y^2 - x^2 = 4\).
This type of simplification is crucial for rewriting a hyperbola's equation into a convenient form, especially when transforming it into its standard version.

Standard Form of Hyperbola

To better understand and work with hyperbolas, it's vital to express their equations in standard form. This form provides clarity about the properties of the hyperbola, such as its vertices and axes. There are two primary standard forms based on the orientation of the hyperbola:

• Horizontal Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Vertical Hyperbola: \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\)

Each form indicates different major axes direction and ensures the magnitude of coefficients aligns with the geometric nature of the hyperbola. In the exercise, \(\frac{y^2}{\frac{1}{4}} - \frac{x^2}{4} = 1\), the constants found as \(b^2 = \frac{1}{4}\) and \(a^2 = 4\) match a vertical hyperbola. Recognizing this form helps in visualizing the graph's orientation.

Mathematical Expansion

The process of expanding mathematical expressions is instrumental in simplifying problems, such as the equation for a hyperbola. Expansion particularly deals with rewriting products of binomials in a straightforward form, simplifying the original equation. For example, the expression

• \((4y - x)(4y + x) = 16y^2 - x^2\)

involves using the difference of squares identity. Identifying and utilizing such identities is critical for further mathematical manipulation and simplification in conic sections. This conversion allows the equation to be more easily adjusted to its standard form, enabling easier graphical interpretation and further algebraic tasks.

Conic Sections

Conic sections represent the curves formed by the intersection of a plane with a double cone. These include ellipses, circles, parabolas, and hyperbolas, each with unique properties and equations. The hyperbola stands out for having two disconnected parts or "branches," unlike other conic sections.
Conic sections can be identified by their equations and are often studied in mathematics due to their prevalence and applications in various scientific fields. The transformation of a hyperbola's equation into standard form relates to understanding its key characteristics such as:

• Center
• Vertices
• Axes
• Foci

Learning about each type of conic section, including hyperbolas, requires comprehending their distinct standard forms, which help in properly visualizing and analyzing their graphs.