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Magazine Chapter 11: Problem 5
An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$
Short Answer
Expert verified
Center: (2,1); Vertices: (-1,1) and (5,1); Foci: (2卤鈭�5,1). Major Axis: 6; Minor Axis: 4.
Step by step solution
01
Identify the Form and Parameters of the Ellipse
The equation of the ellipse is given as \(\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1\). This is in the standard form \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\). Here, \((h,k)\) is the center of the ellipse, \(a^{2}=9\), and \(b^{2}=4\).
02
Determine the Center of the Ellipse
From the standard form, \(h=2\) and \(k=1\). Thus, the center of the ellipse is \((2,1)\).
03
Find the Vertices of the Ellipse
Since \(a^{2}=9\), we have \(a=3\) and the major axis is along the x-axis. The vertices are at \((h\pm a, k) = (2\pm 3, 1) = (-1, 1)\) and \((5, 1)\).
04
Calculate the Foci of the Ellipse
The distance from the center to each focus is \(c\), where \(c = \sqrt{a^{2} - b^{2}} = \sqrt{9 - 4} = \sqrt{5}\). The foci are at \((h\pm c, k) = (2\pm \sqrt{5}, 1) = (2-\sqrt{5}, 1)\) and \((2+\sqrt{5}, 1)\).
05
Determine the Lengths of the Axes
The length of the major axis is \(2a = 2 \times 3 = 6\). The length of the minor axis is \(2b = 2 \times 2 = 4\).
06
Sketch the Graph of the Ellipse
On the Cartesian plane, plot the center at \((2,1)\). Mark the vertices at \((-1, 1)\) and \((5, 1)\), and the endpoints of the minor axis at \((2, -1)\) and \((2, 3)\). Draw a smooth curve connecting these points to complete the ellipse.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Center of an Ellipse
The center of an ellipse is a crucial reference point for understanding its geometric positioning. In the standard form of the ellipse equation, \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the center is represented by the coordinates \((h, k)\). This is where the ellipse is perfectly balanced in both dimensions.
In our example, the ellipse equation \( \frac{(x-2)^2}{9} + \frac{(y-1)^2}{4} = 1 \) shows \( h = 2 \) and \( k = 1 \), making the center \( (2, 1) \).
In our example, the ellipse equation \( \frac{(x-2)^2}{9} + \frac{(y-1)^2}{4} = 1 \) shows \( h = 2 \) and \( k = 1 \), making the center \( (2, 1) \).
- Location: The center is always in the middle of the ellipse, from where all other measurements like vertices and axes are determined.
- Role: It acts as the origin for the ellipse's coordinate system.
Defining the Vertices of an Ellipse
Vertices are the points on an ellipse where it is widest in any given direction. More technically, they are at the ends of the major axis. To determine these points, you focus on either the horizontal or vertical component, depending on which axis is the major one. In the ellipse equation \( \frac{(x-2)^2}{9} + \frac{(y-1)^2}{4} = 1 \), the major axis lies along the x-axis since \( a^2 = 9 > b^2 = 4 \).
The vertices are calculated as \( (h \pm a, k) \). Here, \( a = \sqrt{9} = 3 \), so the vertices are at \( (2 \pm 3, 1) = (-1, 1) \) and \( (5, 1) \).
The vertices are calculated as \( (h \pm a, k) \). Here, \( a = \sqrt{9} = 3 \), so the vertices are at \( (2 \pm 3, 1) = (-1, 1) \) and \( (5, 1) \).
- Position: They lie on the same line parallel to the major axis, mirroring each other across the center.
- Importance: Vertices define the maximum stretch or length of the ellipse, critical for its sketch and structural understanding.
Exploring the Foci of an Ellipse
The foci (plural of focus) of an ellipse have a significant mathematical property: the sum of distances from any point on the ellipse to the two foci is constant. These points lie along the major axis. Determining the foci involves a specific formula where \( c^2 = a^2 - b^2 \). In our example, \( a^2 = 9 \) and \( b^2 = 4 \), giving \( c = \sqrt{9 - 4} = \sqrt{5} \).
The foci are placed at \( (h \pm c, k) \). For \( c = \sqrt{5} \), they are \( (2\pm\sqrt{5}, 1) = (2-\sqrt{5}, 1) \) and \( (2+\sqrt{5}, 1) \).
The foci are placed at \( (h \pm c, k) \). For \( c = \sqrt{5} \), they are \( (2\pm\sqrt{5}, 1) = (2-\sqrt{5}, 1) \) and \( (2+\sqrt{5}, 1) \).
- Placement: They are inside the ellipse, shaping its rounded nature.
- Significance: Foci are essential in definitions and properties of ellipses in conic sections.
Understanding Major and Minor Axes
The axes of an ellipse define its elongation and span. They provide an understanding of the ellipse's dimensions:
They help put the geometric shape into context, showing how the ellipse is stretched in horizontal and vertical directions. Additionally, knowing the lengths can assist with further analysis and calculations, such as determining the area or parameter.
- Major Axis: The longest diameter of the ellipse, passing through both the foci and the center. It is found by \( 2a \). In our example, the major axis length is the horizontal distance, \( 2 \times 3 = 6 \).
- Minor Axis: The shortest diameter, perpendicular to the major axis at the center. Calculated as \( 2b \). For this equation, the minor axis length is \( 2 \times 2 = 4 \).
They help put the geometric shape into context, showing how the ellipse is stretched in horizontal and vertical directions. Additionally, knowing the lengths can assist with further analysis and calculations, such as determining the area or parameter.
