/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 \(x^{2}-4 y^{2}=16\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 10

\(x^{2}-4 y^{2}=16\)

Short Answer

Expert verified
The equation \(x^{2} - 4 y^{2} = 16\) represents a hyperbola.

Step by step solution

01

- Identify the given equation

The given equation is a form of a two-variable quadratic equation: \(x^{2} - 4 y^{2} = 16\).
02

- Recognize the standard form

This equation can be recognized as a hyperbola in its standard form which generally is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
03

- Rewrite the given equation in terms of its standard form

Divide every term in the equation \(x^{2} - 4y^{2} = 16\) by 16 to match the standard form of a hyperbola. This yields: \(\frac{x^2}{16} - \frac{4y^2}{16} = 1\).
04

- Simplify the equation

Simplify the terms in the rewritten equation: \(\frac{x^2}{16} - \frac{y^2}{4} = 1\). Thus, we get the equation in standard hyperbola form: \(\frac{x^2}{16} - \frac{y^2}{4} = 1\).

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Key Concepts

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

Quadratic Equations
A quadratic equation is a type of polynomial equation of the form ewline ewline \[ ax^2 + bx + c = 0 \] ewline where \(a\), \(b\), and \(c\) are constants with \(a e 0\). These equations can come in different forms, such as: ewline
  • Standard Form: \(ax^2 + bx + c = 0\)
  • Factored Form: \(a(x - r_1)(x - r_2) = 0\)
  • Vertex Form: \(a(x - h)^2 + k = 0\)
ewline Quadratic equations are commonly solved using methods like factoring, completing the square, or using the quadratic formula: ewline ewline \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] ewline In the context of our exercise, we deal with a specific type known as a quadratic equation in two variables, typically written as \(Ax^2 + By^2 + C = 0\), which can describe conic sections such as circles, ellipses, parabolas, and hyperbolas.
Standard Form
The standard form of a hyperbola is given by: ewline ewline \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] ewline Where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. This form helps in easily identifying and graphing hyperbolas. To transform an equation into standard form, follow these steps: ewline
  • Isolate all terms on one side of the equation.
  • Factor and simplify the equation to match the standard form. Divide each term by the constant term on the right side to make it equal to 1.
In our exercise, the given equation \(x^2 - 4y^2 = 16\) is divided by 16 to transform it to standard form: ewline ewline \[ \frac{x^2}{16} - \frac{4y^2}{16} = 1 \] ewline This simplifies to: ewline ewline \[ \frac{x^2}{16} - \frac{y^2}{4} = 1 \] ewline Recognizing the standard form helps us understand its geometric representation as a hyperbola.
Hyperbola
A hyperbola is a type of conic section that appears when a plane cuts through both nappes of a double cone. It consists of two disconnected curves called branches. The general equation of a hyperbola in its standard form is: ewline ewline \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] ewline Key features of a hyperbola include: ewline
  • Center: The midpoint between the two vertices of the hyperbola.
  • Vertices: The points where the hyperbola intersects its transverse axis.
  • Foci: Two fixed points located inside each branch of the hyperbola.
  • Asymptotes: Lines that the hyperbola approaches but never touches.
For the exercise equation \( \frac{x^2}{16} - \frac{y^2}{4} = 1 \): ewline
  • \(a = 4\) (since \(a^2 = 16\))
  • \(b = 2\) (since \(b^2 = 4\))
These parameters help in graphing and further understanding the hyperbola's properties. Hyperbolas are useful in many applications, including physics, astronomy, and navigation systems.

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Most popular questions from this chapter

Prove the given property if \(a\) is any positive. number and \(x\) and \(y\) are any real numbers. $$ a^{x} \div a^{y}=a^{x-y} $$

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\(\int \frac{d x}{x \ln x}\)

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