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Chapter 12: Problem 17

Identify the graph of each equation as a parabola, circle, ellipse, or hyperbola, and then sketch the graph. $$ 4 x^{2}+y^{2}=16 $$

Short Answer

Expert verified
Equation represents an ellipse centered at origin with axes lengths 4 and 8.

Step by step solution

01

Identify the Standard Form

Rewrite the given equation in a standard conic form. The equation is \( 4x^{2} + y^{2} = 16 \).
02

Divide by the Constant

Divide the entire equation by 16 to normalize the equation. This gives \( \frac{4x^{2}}{16} + \frac{y^{2}}{16} = 1 \).
03

Simplify the Equation

Simplify the fractions to identify the conic section. The equation simplifies to \( \frac{x^{2}}{4} + \frac{y^{2}}{16} = 1 \).
04

Identify the Conic Section

Compare the simplified equation \( \frac{x^{2}}{4} + \frac{y^{2}}{16} = 1 \) with the standard forms of conic sections. The equation matches the form of an ellipse \( \frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 1 \), where \( a = 2 \) and \( b = 4 \).
05

Sketch the Graph

Since it is an ellipse centered at the origin with semi-major axis along the y-axis (\( b = 4 \)) and semi-minor axis along the x-axis (\( a = 2 \)), plot the points at (\( \pm 2, 0 \)) and (\( 0, \pm 4 \)), then sketch the ellipse passing through these points.

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Key Concepts

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

Ellipse
An ellipse is a type of conic section that looks like an elongated circle. It has two axes: the major axis and the minor axis. The major axis is the longest distance across the ellipse, and the minor axis is the shortest. An important property of an ellipse is its foci (plural of focus). Every point on the ellipse has the same total distance from these two foci.To write the equation of an ellipse, you'll need the lengths of the semi-major axis and the semi-minor axis.The standard form of the equation of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a \) is the length of the semi-major axis, and \( b \) is the length of the semi-minor axis.
Graphing Conic Sections
Graphing conic sections requires identifying key characteristics such as the center, vertices, and axes. Each type of conic section鈥攑arabola, circle, ellipse, and hyperbola鈥攈as distinct features that can be used to graph it accurately.
  • For a parabola, identify the vertex and the focus.
  • For a circle, identify the center and the radius.
  • For an ellipse, identify the center, lengths of the major and minor axes, and the foci.
  • For a hyperbola, identify the center, vertices, and asymptotes.
Use these key points to plot the graph. For ellipses, plot the endpoints of the major and minor axes, then sketch the smooth curve passing through these points.Try to maintain symmetry, as ellipses are symmetrical about both their major and minor axes.
Standard Form Equations
Standard form equations make it easier to identify and graph conic sections. Let's discuss the standard forms for each type of conic section:
  • Parabola: \((x-h)^2 = 4p(y-k)\)
  • Circle: \((x-h)^2 + (y-k)^2 = r^2\)
  • Ellipse: \((x-h)^2 / a^2 + (y-k)^2 / b^2 = 1\)
  • Hyperbola: \((x-h)^2 / a^2 - (y-k)^2 / b^2 = 1\)
Each type of conic section can be transformed into its standard form, which allows you to identify key properties like vertices, axes, and foci easily.For example, the given equation \( 4x^2 + y^2 = 16 \) was rewritten as \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \), identifying it as an ellipse with center at the origin, semi-major axis of 4 along the y-axis, and semi-minor axis of 2 along the x-axis.

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Most popular questions from this chapter

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