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Chapter 11: Problem 43

Find the center, foci, and vertices of each ellipse. Graph each equation. $$\frac{(x-3)^{2}}{4}+\frac{(y+1)^{2}}{9}=1$$

Short Answer

Expert verified
The center is \(3, -1\), vertices are \((3, -4)\) and \((3, 2)\), foci are at \((3, -1 \pm \sqrt{5})\).

Step by step solution

01

Identify the form of the ellipse equation

The given equation is \[\frac{(x-3)^{2}}{4}+\frac{(y+1)^{2}}{9}=1\]. This is in the 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, \(a^{2}\) is the denominator under the \(x\) term, and \(b^{2}\) is the denominator under the \(y\) term.
02

Determine the center of the ellipse

By comparing the given equation \[\frac{(x-3)^{2}}{4}+\frac{(y+1)^{2}}{9}=1\] to the standard form \[\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\], it can be seen that \(h = 3\) and \(k = -1\). Therefore, the center of the ellipse is at the point \((3, -1)\).
03

Identify the values of \(a\) and \(b\)

In the given equation, \(a^{2} = 4\) and \(b^{2} = 9\). Calculating \(a\) and \(b\), we get \(a = 2\) and \(b = 3\).
04

Find the vertices of the ellipse

The vertices are \(a\) units away from the center along the major axis. Since \(b = 3\) is greater than \(a = 2\), the major axis is vertical. The vertices are \((3, -1 \pm 3)\) which results in the points \((3, -4)\) and \((3, 2)\).
05

Find the foci of the ellipse

The distance from the center to each focus is given by \(c\), where \(c = \sqrt{b^{2} - a^{2}}\). Substituting the values, we have \[c = \sqrt{9 - 4} = \sqrt{5}\]. Thus, the foci are \((3, -1 \pm \sqrt{5})\). This results in \((3, -1 - \sqrt{5})\) and \((3, -1 + \sqrt{5})\).
06

Graph the ellipse

To graph the ellipse, plot the center at \((3, -1)\), vertices at \((3, -4)\) and \((3, 2)\), and the foci at approximately \((3, -3.24)\) and \((3, 1.24)\). Draw the ellipse through these points ensuring the longer axis is vertical as determined from the values of \(a\) and \(b\).

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Key Concepts

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

center of ellipse
The center of an ellipse is a crucial point that represents the midpoint of the ellipse.
It can be found directly from the standard form of the ellipse equation.
For an equation given in the form \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\), the center of the ellipse is at the point \((h, k)\).

In our example, the equation is \(\frac{(x-3)^{2}}{4}+\frac{(y+1)^{2}}{9}=1\).
Here, by comparing it to the standard form, we identify \((h, k)\) as \(3\) and \(-1\) respectively.
Thus, the center of the ellipse is at \((3, -1)\).
This point serves as the starting point for locating the vertices and foci of the ellipse.
vertices of ellipse
Vertices are key points on an ellipse located at the ends of the major axis.
The equation of our ellipse is \(\frac{(x-3)^{2}}{4}+\frac{(y+1)^{2}}{9}=1\).
We derive \(\text{a} = 2\) and \(\text{b} = 3\) from the denominators of the ellipse equation.

Since \(\text{b} = 3\) is larger than \(\text{a} = 2\), the major axis is vertical.
Thus, the vertices will be \(\text{b}\) units up and down from the center point \((3, -1)\).
This results in the vertices at \((3, -4)\) and \((3, 2)\).
The vertices give us essential information about the extremities of the ellipse.
foci of ellipse
The foci (singular: focus) are points located along the major axis of an ellipse.
They are crucial in defining the shape of the ellipse.
For an ellipse, the distance from the center to each focus is given by \(c\), where \(c = \sqrt{b^{2} - a^{2}}\).

In our equation \(a^{2} = 4\) and \(b^{2} = 9\). Thus, \(c = \sqrt{9 - 4} = \sqrt{5}\).
Substituting \(c\) in, the foci are \((3, -1 - \sqrt{5})\) and \((3, -1 + \sqrt{5})\).
These positions are significant because any point on the ellipse is such that the sum of its distances from the two foci is constant.
graphing ellipse
Graphing an ellipse involves plotting its important points, such as the center, vertices, and foci.
Begin by marking the center \(3, -1\).
Next, locate the vertices at \((3, -4)\) and \(3, 2\).

Identify the foci at approximately \((3, -3.24)\) and \((3, 1.24)\).
It's crucial to connect these dots smoothly, ensuring that the ellipse maintains the characteristic elongated shape along the vertical axis.
This visual representation helps in understanding how the equation defines the shape and orientation of the ellipse.

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Most popular questions from this chapter

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