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Magazine Chapter 10: Problem 38
For the following exercises, graph the given ellipses, noting center, vertices, and foci. $$ \frac{x^{2}}{2}+\frac{(y+1)^{2}}{5}=1 $$
Short Answer
Expert verified
The ellipse is centered at (0, -1) with vertices (0, -1 卤 鈭�5) and foci (0, -1 卤 鈭�3).
Step by step solution
01
Convert to Standard Form
The given equation of the ellipse is \(\frac{x^{2}}{2}+\frac{(y+1)^{2}}{5}=1\). This is already in standard form of an ellipse, which is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \( (h,k) \) is the center.
02
Identify the Center
From the equation, \( x^2 \) implies \( h = 0 \) and \( (y+1)^2 \) implies \( k = -1 \). Thus, the center of the ellipse is at \((0, -1)\).
03
Determine Axis Lengths
In the equation \(\frac{x^{2}}{2}+\frac{(y+1)^{2}}{5}=1\), \( a^2 = 5 \) and \( b^2 = 2 \). \( a^2 \) corresponds to the term with \( y \), indicating the major axis is along the y-direction. Hence, \( a = \sqrt{5} \) and \( b = \sqrt{2} \).
04
Locate the Vertices
Since the major axis is vertical, the vertices are \( k \pm a \). Thus, they are at \((0, -1 + \sqrt{5})\) and \((0, -1 - \sqrt{5})\).
05
Find the Foci
The distance to each focus from the center is given by \( c \), where \( c = \sqrt{a^2 - b^2} = \sqrt{5 - 2} = \sqrt{3} \). The foci are on the major axis: \((0, -1 + \sqrt{3})\) and \((0, -1 - \sqrt{3})\).
06
Graph the Ellipse
Plot the center at \((0, -1)\), vertices at \((0, -1 + \sqrt{5})\) and \((0, -1 - \sqrt{5})\), and foci at \((0, -1 + \sqrt{3})\) and \((0, -1 - \sqrt{3})\). The ellipse is vertically elongated due to the larger \( a^2 \). Draw the ellipse around these points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Standard Form of an Ellipse
The equation provided \( \frac{x^2}{2} + \frac{(y+1)^2}{5} = 1 \) is already in the standard form of an ellipse. The general form of an ellipse is expressed as: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] This formula is essential because it helps identify important features of the ellipse, such as the center, vertices, and foci.
- \( h \) and \( k \) are the coordinates of the center.
- \( a^2 \) and \( b^2 \) are the squares of the semi-major and semi-minor axes, respectively.
Center of an Ellipse
Finding the center of an ellipse is a pivotal step in understanding its geometry. The center is directly derived from the standard form equation. For the ellipse \( \frac{x^2}{2} + \frac{(y+1)^2}{5} = 1 \), we observe the terms:
- The \( x^2 \) term with no horizontal shift indicates \( h = 0 \).
- The \( y+1 \) term implies a vertical shift, so \( k = -1 \).
Vertices of an Ellipse
Vertices are key points on an ellipse, located at the endpoints of its major axis. In our equation, since \( a^2 = 5 \) and \( b^2 = 2 \), the ellipse is vertically oriented. Here's how to find the vertices:
- Calculate \( a \) by finding \( \sqrt{5} \).
- Use the center \( (0, -1) \), and since the major axis is vertical, adjust the \( y \)-coordinate:
- Vertices are located at \( (0, -1 + \sqrt{5}) \) and \( (0, -1 - \sqrt{5}) \).
Foci of an Ellipse
The foci are two central points located along the major axis of the ellipse, and they play a significant role in its shape. To find the foci, we calculate the value \( c \): \[ c = \sqrt{a^2 - b^2} = \sqrt{5 - 2} = \sqrt{3} \]
- With \( c \) known, determine the foci along the vertical major axis from the center \( (0, -1) \).
- The foci are at \( (0, -1 + \sqrt{3}) \) and \( (0, -1 - \sqrt{3}) \).
Graphing Ellipses
Graphing an ellipse involves combining all these found elements to create an accurate sketch. Steps to graph the given ellipse:
- Begin by marking the center at \( (0, -1) \).
- Plot the vertices at \( (0, -1 + \sqrt{5}) \) and \( (0, -1 - \sqrt{5}) \) along the vertical major axis.
- Identify the foci at \( (0, -1 + \sqrt{3}) \) and \( (0, -1 - \sqrt{3}) \).
- Draw the ellipse around these points, ensuring it is vertically elongated.
