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Chapter 10: Problem 27

Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{4}+y^{2}=1$$

Short Answer

Expert verified
Answer: For the given ellipse equation, the lengths of the major and minor axes are 2 and 1, respectively. The coordinates of the vertices are $$(2, 0), (-2, 0), (0, 1), (0, -1)$$, and the coordinates of the foci are $$(-\sqrt{3}, 0)$$ and $$(\sqrt{3}, 0)$$. To sketch the ellipse, draw the major and minor axes intersecting at the origin, plot the vertices' and foci's coordinates, and connect the points to form the ellipse.

Step by step solution

01

Identify the ellipse equation format

The general equation of an ellipse centered at the origin is given by $$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$, where a and b are the lengths of the major and minor axes, respectively. Compare the given equation $$\frac{x^2}{4} + y^2=1$$ with the standard ellipse equation, we notice that $$a^2=4$$ and $$b^2=1$$.
02

Find the lengths of the major and minor axes

Now that we know the values of $$a^2$$ and $$b^2$$, we can find the lengths of the major (a) and minor (b) axes: $$a = \sqrt{a^2} = \sqrt{4} = 2$$ $$b = \sqrt{b^2} = \sqrt{1} = 1$$
03

Determine the coordinates of the vertices

The coordinates of the vertices of an ellipse centered at the origin are given by $$(\pm a, 0)$$ and $$(0, \pm b)$$. In this case, the coordinates for the vertices are: $$(\pm 2, 0)$$ and $$(0, \pm 1)$$. These coordinates are: $$(2, 0), (-2, 0), (0, 1), (0, -1)$$.
04

Calculate the coordinates of the foci

The coordinates of the foci for an ellipse centered at the origin are determined by the distance (c) from the center to the foci. In this case, the foci are along the x-axis, and the distance c can be found using the formula: $$c = \sqrt{a^2 - b^2}$$ $$c = \sqrt{4 - 1} = \sqrt{3}$$ The coordinates of the foci are $$(\pm c, 0)$$. So, the coordinates of the foci for our ellipse are $$(\pm \sqrt{3}, 0)$$. Which means the coordinates of the foci are: $$(-\sqrt{3}, 0)$$ and $$(\sqrt{3}, 0)$$.
05

Sketch the ellipse and label the coordinates

Now that we have all the necessary information, we can sketch the ellipse. First, draw the major and minor axes which intersect at the origin (0, 0). Next, plot the vertices coordinates $$(2, 0), (-2, 0), (0, 1), (0, -1)$$. And finally, plot the foci coordinates $$(-\sqrt{3}, 0)$$ and $$(\sqrt{3}, 0)$$. Connect the points and carefully sketch the ellipse.
06

Verify your work with a graphing utility

To check our work, we can use a graphing utility such as GeoGebra or Desmos to input the ellipse equation $$\frac{x^2}{4} + y^2 = 1$$ and confirm that the graph aligns with our sketch and the coordinates of the vertices and foci that we found.

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Key Concepts

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

Major Axis
In an ellipse, the major axis is its longest diameter, stretching from one vertex of the ellipse to the other. It spans across the widest part of the ellipse.
The length of the major axis is determined based on the equation of the ellipse. For a standard ellipse equation \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] if \(a > b\), then the length of the major axis will be \(2a\). Here, 'a' represents the semi-major axis, which is the distance from the center to a vertex along the most extended axis.
In our example equation \[ \frac{x^2}{4} + y^2 = 1 \], \(a^2 = 4\) implies \(a = 2\), so the length of the major axis is \(2 \times 2 = 4\).
This axis is critical as it indicates the longest distance across the ellipse, passing through the center. It's essential for sketching proper, proportional drawings and understanding the ellipse's orientation.
Minor Axis
The minor axis of an ellipse is the shortest diameter that runs perpendicular to the major axis. It crosses the ellipse鈥檚 center, forming a shorter cross-section.
Based on the same standard ellipse equation\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]鈥� if \(b < a\), it represents the length of the minor axis as \(2b\). Here, 'b' is the semi-minor axis, which measures the distance from the center to the ellipse鈥檚 edge along the shortest axis.
In the problem at hand, \(b^2 = 1\) gives \(b = 1\), and so the minor axis measures \(2 \times 1 = 2\).
This component of the ellipse is necessary for shaping its overall proportion and balance. It helps in defining the ellipse's compactness and is fundamental when visualizing the structure accurately.
Foci
The foci are two fixed points on the interior of an ellipse used to define its shape. Each point is equidistant from the center and located along the major axis.
To find the foci鈥檚 distance from the ellipse center, we apply the formula:\[ c = \sqrt{a^2 - b^2} \]where '\(c\)' is the focal distance.In our exercise:\[ a^2 = 4,\, b^2 = 1,\, so \quad c = \sqrt{4 - 1} = \sqrt{3} \]These foci are positioned at \((\pm \sqrt{3}, 0)\), revealing their presence along the horizontal axis.
The foci play a crucial role in the ellipse鈥檚 geometry. An interesting feature is that for any point on the ellipse, the sum of the distances to the foci remains constant. Understanding the foci's roles helps in comprehending how an ellipse is not merely a circle but stretched in a particular direction.

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Most popular questions from this chapter

An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations $$x(\theta)=a \cos \theta+\cos n \theta, y(\theta)=a \sin \theta+\sin n \theta.$$ The distance from the moon to the planet is taken to be \(1,\) the distance from the planet to the Sun is \(a,\) and \(n\) is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of \(n\) produce loops for a fixed value of \(a\) a. \(a=4, n=3\) b. \(a=4, n=4 \) c. \(a=4, n=5\)

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Consider the parametric equations $$ x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t $$ where \(a, b, c,\) and \(d\) are real numbers. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K,\) where \(A, B, C,\) and \(K\) are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the \(x\) - and \(y\) -axes provided \(a b+c d=0\) c. Show that the equations describe a circle provided \(a b+c d=0\) and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\)

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