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Chapter 9: Problem 65

Find the asymptotes of the graph of the hyperbola given by \(9 y^{2}-16 x^{2}-36 y-32 x-124=0\)

Short Answer

Expert verified
Asymptotes: \(y = \frac{4}{3}x + \frac{10}{3}\) and \(y = -\frac{4}{3}x - \frac{2}{3}\).

Step by step solution

01

Rewrite the Equation

Begin by rewriting the given equation in a form that closely resembles the standard form of a hyperbola. The given equation is:\[9 y^{2} - 16 x^{2} - 36 y - 32 x - 124 = 0\]Rearrange it to group the terms involving \(y\) and \(x\):\[9(y^2 - 4y) - 16(x^2 + 2x) = 124\]
02

Complete the Square

Complete the square for both \(y\) and \(x\) terms within the rearranged equation.- For the \(y\) terms, \(y^2 - 4y\), add and subtract \(4\) (\((4/2)^2\)): \[(y^2 - 4y + 4 - 4)\]- For the \(x\) terms, \(x^2 + 2x\), add and subtract \(1\) (\((2/2)^2\)): \[(x^2 + 2x + 1 - 1)\]Substituting back, we have:\[9[(y-2)^2 - 4] - 16[(x+1)^2 - 1] = 124\]
03

Simplify the Equation

Simplify and reorganize the equation by distributing the constants and balancing both sides:\[9(y-2)^2 - 36 - 16(x+1)^2 + 16 = 124\]Combine constants:\[9(y-2)^2 - 16(x+1)^2 = 144\]
04

Divide to Identify the Standard Form

Divide through by 144, the constant on the right side, to achieve the standard form of a hyperbola:\[\frac{(y-2)^2}{16} - \frac{(x+1)^2}{9} = 1\]Thus, we identify that the hyperbola is oriented vertically, since \(y\) is the positive term.
05

Determine the Asymptote Equations

For a hyperbola in the form \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\), the asymptotes are given by:\[y - k = \pm\frac{a}{b}(x - h)\]Here, \(a = 4\), \(b = 3\), \(h = -1\), and \(k = 2\), yielding:\[(y - 2) = \pm\frac{4}{3}(x + 1)\]Therefore, the asymptotes are:\[y = \frac{4}{3}x + \frac{10}{3}\] and \[y = -\frac{4}{3}x - \frac{2}{3}\]

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Key Concepts

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

Hyperbola
A hyperbola is a type of conic section, just like circles and ellipses, but with distinct characteristics. It is formed by the intersection of a double cone and a plane. The typical appearance of a hyperbola resembles two opposite-facing curves or arms. Understanding the basics of this shape is key to solving problems related to it.

Some features of a hyperbola include:
  • Two branches or curves facing away from each other.
  • A center point, from which distances to certain points (known as foci) are calculated.
  • Asymptotes - lines that the hyperbola approaches, but never actually touches.
Hyperbolas are used widely in various real-world applications, such as in satellite communications and radio signals.
Understanding its algebraic equation allows us to graph these curves accurately.
Completing the Square
Completing the square is a mathematical technique used to reshape quadratic equations into a form that is easier to analyze. It is an essential step when working with the equations of hyperbolas, ellipses, and circles, as it helps transform the equation into a standard form. Here鈥檚 how completing the square greatly aids in analyzing hyperbolas:

  • Firstly, it allows rearranging certain quadratic parts of the equation to form perfect square trinomials.
  • By adding and subtracting specific values, we can condense various terms into {\( (y - k)^2 \) and \( (x - h)^2 \) - these represent shifted squares that align more naturally with graph axes.
  • This process reveals the shape, position, and orientation of the hyperbola.
For completing the square, remember to balance the equation by adjusting both sides, ensuring it remains equivalent. It might seem like a complex process at first, but it simplifies graphing and identifying key properties of the curve.
Standard Form of a Hyperbola
The standard form of a hyperbola's equation provides crucial insights about its characteristics and graph. For hyperbolas oriented vertically, the standard form is \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\). Conversely, for those oriented horizontally, it appears as \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\). The parameters in these equations significantly shape our understanding.

  • The \( (h, k) \) components represent the center of the hyperbola, or the point around which it's symmetrical.
  • \( a \) and \( b \) define the slopes of the asymptotes, which are pivotal in sketching the graph accurately.
  • The signs in front of the fractions determine the orientation of the hyperbola.
Once in standard form, the asymptotes' equations can easily be derived, serving as a reference for sketching the graph. Balancing the equation by dividing allows you to uncover this form and fully utilize the hyperbola's properties.

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Most popular questions from this chapter

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