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Chapter 10: Problem 90

If \(D \neq 0\) or \(E \neq 0,\) then the graph of \(y^{2}-x^{2}+D x+E y=0\) is a hyperbola.

Short Answer

Expert verified
The given equation \(y^{2}-x^{2}+D x+E y =0\) represents a hyperbola where \(D \neq 0\) or \(E \neq 0\).

Step by step solution

01

Identification

Identify the equation given. In this case, the general equation is \(y^{2}-x^{2}+D x+E y=0\) where \(D \neq 0\) or \(E \neq 0\).
02

Rearrangement

Rearrange the equation to make it look more like a standard hyperbola equation. If D and E are not both zero, it implies that either x or y is not only squared but also has a non-zero multiple, therefore rearrange to \(y^{2}-E y - x^{2} +Dx =0\) .
03

Completing the square

Complete the square for the rearranged equation, \( (y^2-Ey+ (E/2)^2) - (x^2-Dx+ (D/2)^2) = (E/2)^2 - (D/2)^2\), which is reduced to \((y-E/2)^2 - (x-D/2)^2 = (E/2)^2 - (D/2)^2\) .
04

Identify the hyperbola

Note that this equation is in the standard form of a hyperbola equation \((y-k)^2/b^2 - (x-h)^2/a^2 = 1\), where \(a^2 = (E/2)^2\), \(b^2 = (D/2)^2\), \(h = D/2\), and \(k = E/2\). Therefore, the given equation indeed represents a hyperbola.

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Key Concepts

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

Conic Sections
Conic sections are curves obtained by intersecting a double cone with a plane. Each type of conic section has a unique shape and equation. The most common conic sections are:
  • Circles
  • Ellipses
  • Parabolas
  • Hyperbolas
Among these, hyperbolas are formed when the intersecting plane cuts both halves of the cone, creating two distinct curves that open in opposite directions.
Hyperbolas have equations of the form:\[\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\]where \(h\) and \(k\) represent the center of the hyperbola. This standard form allows us to easily identify and graph a hyperbola.
Completing the Square
Completing the square is a crucial technique used in algebra to simplify and solve quadratic equations. Here鈥檚 how it works:
  • Identify a quadratic expression: \(ax^2 + bx + c\).
  • Divide \(b\) by 2 and square the result.
  • Add and subtract this square inside the equation.
For the equation \(y^2 - Ey\), completing the square involves rewriting it as:\[(y - \frac{E}{2})^2 - (\frac{E}{2})^2\]Completing the square helps us convert a general quadratic equation into its standard form, making it easier to identify its graph as a hyperbola in this case.
Equation Rearrangement
Rearranging equations is a strategy to make them easier to work with. It involves changing the order and grouping of terms without altering the equation鈥檚 meaning. For converting an equation into a form that highlights its conic section type, rearranging is vital.
In the given problem, the hyperbola equation \(y^2 - x^2 + Dx + Ey = 0\) is rearranged to:\[(y^2 - Ey) - (x^2 - Dx) = 0\]This layout is more conducive to completing the square. Careful rearrangement reveals the core structure of the equation, making the process of turning it into the standard hyperbola form straightforward.

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Most popular questions from this chapter

Identifying a Conic In Exercises \(23-26,\) use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. $$r=\frac{6}{6+7 \cos \theta}$$

Identifying a Conic In Exercises \(23-26,\) use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. $$r=\frac{3}{-4+2 \sin \theta}$$

EXPLORING CONCEPTS Using Different Methods In Exercises 71 and \(72,\) (a) sketch the graph of the polar equation, (b) determine the interval that traces the graph only once, (c) find the area of the region bounded by the graph using a geometric formula, and (d) find the area of the region bounded by the graph using integration. $$r=5 \sin \theta$$

Area and Arc Length in Polar Coordinates Spiral of Archimedes The curve represented by the equation \(r=a \theta,\) where \(a\) is a constant, is called the spiral of Archimedes. (a) Use a graphing utility to graph \(r=\theta,\) where \(\theta \geq 0 .\) What happens to the graph of \(r=a \theta\) as \(a\) increases? What happens if \(\theta \leq 0 ?\) (b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0),\) where the curve crosses the polar axis. (c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\) (d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\)

Finding the Area of a Surface of Revolution In Exercises \(65-68\) , find the area of the surface formed by revolving the polar equation over the given interval about the given line. $$\begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\ {r=a \cos \theta} & {0 \leq \theta \leq \frac{\pi}{2}}\end{array} \begin{array}{ll}{\text { Axis of Revolution }} \\\ {\theta=\frac{\pi}{2}}\end{array}$$
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