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Chapter 10: Problem 21

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}-4 y^{2}-2 x+16 y=20$$

Short Answer

Expert verified
The equation represents a hyperbola.

Step by step solution

01

Rearrange the equation

Start by rearranging the given equation to organize the terms: \ \[ x^2 - 2x - 4y^2 + 16y = 20. \]
02

Complete the square for x terms

For the x terms, take half of the coefficient of x, square it, and add and subtract this value inside the expression. The coefficient of x is -2: \ \[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4}. \] \ Add and subtract 1/4 inside the equation: \ \[ (x^2 - 2x + \frac{1}{4}) - \frac{1}{4}. \] This simplifies to \( (x-1)^2 \).
03

Complete the square for y terms

For the y terms, factor out -4 from \(-4y^2 + 16y\) to make completing the square easier: \ \[ -4(y^2 - 4y) \] \ Take half of the coefficient of y, square it, and add and subtract this value: \ \[ \left(-\frac{4}{2}\right)^2 = 4. \] \ Add 4 within the parentheses and multiply it by -4 outside: \ \[ -4(y^2 - 4y + 4) + 16 = -4(y-2)^2 + 16. \]
04

Rewrite the equation

Substitute back the completed squares into the equation: \ \[ (x-1)^2 - 4(y-2)^2 = 20 + \frac{1}{4} - 16. \] \ Simplify the constants on the right side: \ \[ (x-1)^2 - 4(y-2)^2 = \frac{21}{4} - 16 = \frac{5}{4}. \]
05

Identify the conic section

The standard form for hyperbolas is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( -\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Rewrite as: \ \[ \frac{(x-1)^2}{\frac{5}{4}} - \frac{(y-2)^2}{\frac{5}{16}} = 1 \]. This is a hyperbola!

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Key Concepts

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

Hyperbola
A hyperbola is one of the four types of conic sections, which are curves obtained by intersecting a plane with a double napped cone. Unlike a circle or an ellipse, a hyperbola has two distinct branches. Its principal property is that the difference of the distances from any point on the hyperbola to two fixed points (called foci) is constant. A hyperbola typically has the form:
  • Standard horizontal equation: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
  • Standard vertical equation: \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \)
Here, \((h, k)\) is the center of the hyperbola. The terms \(a^2\) and \(b^2\) determine the shape and orientation of the hyperbola. To sketch or analyze a hyperbola, you need to:
  • Identify the center \((h, k)\).
  • Find the foci using the formula: \( c = \sqrt{a^2 + b^2} \). The foci will be located at \((h \pm c, k)\) for horizontal and \((h, k \pm c)\) for vertical hyperbolas.
  • Determine the vertices by calculating \((h \pm a, k)\) or \((h, k \pm a)\), based on orientation.
  • Find the asymptotes. For a horizontal hyperbola, the asymptotes are given by \( y = k \pm \frac{b}{a}(x-h) \), and vice versa for a vertical hyperbola.
Completing the Square
Completing the square is a mathematical technique used to derive certain expressions into a perfect square trinomial. It is especially useful in transforming quadratic equations. By rewriting an equation in its completed square form, you can easily identify key properties of conic sections, such as the vertex and the axis of symmetry.Here's the step-by-step method for completing the square:
  • First, focus on the quadratic terms (those involving \(x^2\) or \(y^2\)).
  • For a term like \(x^2 + bx\), take half of the coefficient of \(x\) (which is \(b\)), square it, and add and subtract it within the expression.
  • The term \((\frac{b}{2})^2\) completes the square, turning \(x^2 + bx\) into \((x + \frac{b}{2})^2 - (\frac{b}{2})^2\).
Completing the square is incredibly useful in identifying the form of a conic section. For the example provided, this technique helps to recognize the expression as a hyperbola after rearranging and simplifying the equation.
Conic Section Properties
Conic sections are curves that can be formed by intersecting a plane with a cone. There are four main types:
  • Circle: A special case of an ellipse where both foci are at the same point. The equation has the form \(x^2 + y^2 = r^2\), where \(r\) is the radius.
  • Ellipse: An elongated circle with two foci and the constant sum of distances from any point on the ellipse to these foci. The standard form is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
  • Parabola: A conic section where the plane is parallel to a generator of the cone. Characterized by a single focus and a directrix. The standard form is \(y = ax^2 + bx + c\).
  • Hyperbola: Defined by two symmetrical branches. It has the properties explained earlier, and there is a constant difference between the distances from any point to the foci. The standard forms are \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) or \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\).
Properties of each conic section like foci, directrices, eccentricity, and vertices determine the geometry and spatial orientation of these curves. Understanding these properties allows us to analyze and sketch their graphs accurately.

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Most popular questions from this chapter

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}-y^{2}=2 y, \quad \phi=\cos ^{-1} \frac{3}{5}$$

In this section we stated that parametric equations contain more information than just the shape of a curve. Write a short paragraph explaining this statement. Use the following example and your answers to parts (a) and (b) below in your explanation. The position of a particle is given by the parametric equations $$x=\sin t \quad y=\cos t$$ where \(t\) represents time. We know that the shape of the path of the particle is a circle. (a) How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. (b) Does the particle travel clockwise or counterclockwise around the circle? Find parametric equations if the particle moves in the opposite direction around the circle.

A Family of Confocal Conics Conics that share a focus are called confocal. Consider the family of conics that have a focus at \((0,1)\) and a vertex at the origin (see the figure). (a) Find equations of two different ellipses that have these properties. (b) Find equations of two different hyperbolas that have these properties. (c) Explain why only one parabola satisfies these properties. Find its equation. (d) Sketch the conics you found in parts (a), (b), and (c) on the same coordinate axes (for the hyperbolas, sketch the top branches only) (e) How are the ellipses and hyperbolas related to the parabola?

Use a graphing device to graph the conic. $$x^{2}-4 y^{2}+4 x+8 y=0$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}+16=4\left(y^{2}+2 x\right)$$
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