Problem 8
Question
Consider the ellipse given by \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1.\) Is the ellipse elongated or nearly circular?
Step-by-Step Solution
Verified Answer
The ellipse is elongated along the y-axis (since 2 < 8).
1Step 1: Identify Semimajor and Semiminor Axes
In the given equation, \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1\), 2 and 8 are the lengths of semimajor and semiminor axes respectively. So, a is 2 and b is 8.
2Step 2: Compare Values of Semimajor and Semiminor Axes
Compare the values of semimajor and semiminor axes. Here, a is less than b.
Key Concepts
Semimajor AxisSemiminor AxisEquation of an Ellipse
Semimajor Axis
The semimajor axis is one of the two principal axes of an ellipse. This axis is the longest diameter that runs through the center of the ellipse, reaching from one end of the ellipse to the other.
The length of the semimajor axis is often denoted by the letter \( a \). In the equation of an ellipse given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a \) is associated with the larger denominator under \( y^2 \), assuming the orientation of the ellipse is along the y-axis.
In our exercise, the semimajor axis has a length of 8. It is aligned with the y-axis in the standard form of the equation. Since it is the larger of the two axes, it indicates the direction in which the ellipse is elongated. The longer the semimajor axis, the more pronounced the elongation.
The length of the semimajor axis is often denoted by the letter \( a \). In the equation of an ellipse given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a \) is associated with the larger denominator under \( y^2 \), assuming the orientation of the ellipse is along the y-axis.
In our exercise, the semimajor axis has a length of 8. It is aligned with the y-axis in the standard form of the equation. Since it is the larger of the two axes, it indicates the direction in which the ellipse is elongated. The longer the semimajor axis, the more pronounced the elongation.
Semiminor Axis
The semiminor axis is the second principal axis of an ellipse and represents the shortest diameter of the ellipse. Unlike the semimajor axis, the semiminor stretches through the center perpendicularly within the bounds of the ellipse.
This length is represented by \( b \) in the standard equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( b^2 \) is the denominator under \( x^2 \), provided the major axis is aligned vertically.
This length is represented by \( b \) in the standard equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( b^2 \) is the denominator under \( x^2 \), provided the major axis is aligned vertically.
- In our specified ellipse equation, its length is determined to be 2.
- Being less than the semimajor axis, it helps define the oval shape as opposed to a circle.
Equation of an Ellipse
The equation of an ellipse is a quadratic expression in two variables, \( x \) and \( y \), which represents all the points that form the ellipse. It takes the standard form of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the lengths of the semimajor and semiminor axes, respectively.
This equation, \( \frac{x^{2}}{2^{2}} + \frac{y^{2}}{8^{2}} = 1 \), shows the ellipse in standard position with the center at the origin (0,0). The variables \( x^2 \) and \( y^2 \) are normalized by their respective axis lengths, \( a^2 \) and \( b^2 \).
By analyzing this equation, one can quickly identify whether the ellipse is elongated or near circular:
This equation, \( \frac{x^{2}}{2^{2}} + \frac{y^{2}}{8^{2}} = 1 \), shows the ellipse in standard position with the center at the origin (0,0). The variables \( x^2 \) and \( y^2 \) are normalized by their respective axis lengths, \( a^2 \) and \( b^2 \).
By analyzing this equation, one can quickly identify whether the ellipse is elongated or near circular:
- If \( a \) equals \( b \), the ellipse is a circle.
- If \( a \) is not equal to \( b \), as in the given exercise, the ellipse is elongated along the axis with the larger value (here, the semimajor axis).
Other exercises in this chapter
Problem 8
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Consider the parametric equations \(x=\sqrt{t}\) and \(y=2-t.\) (a) Create a table of \(x\) - and \(y\) -values using \(t\)=0,1, 2, 3, and 4. (b) Plot the point
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Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi
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