Problem 17 - College Algebra

Question

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. $$ \frac{(x+5)^{2}}{4}+\frac{(y-7)^{2}}{9}=1 $$

Step-by-Step Solution

Answer

Center: (-5, 7), Major axis endpoints: (-5, 10) and (-5, 4), Minor axis endpoints: (-3, 7) and (-7, 7), Foci: \((-5, 7+\sqrt{5})\) and \((-5, 7-\sqrt{5})\).

Step 1: Identify the Standard Form of the Ellipse Equation

The given equation \( \frac{(x+5)^{2}}{4}+\frac{(y-7)^{2}}{9}=1 \) is in the form \( \frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}=1 \), where \( a^{2} = 9 \) and \( b^{2} = 4 \). Since \( a^{2} > b^{2} \), this is a vertical ellipse.

Step 2: Determine the Center of the Ellipse

The center \((h, k)\) of the ellipse is given by \((-5, 7)\) from the equation components \((x+5)\) and \((y-7)\). This represents a translation of the center from the origin to the point \((-5, 7)\).

Step 3: Calculate the Lengths of the Major and Minor Axes

For a vertical ellipse, the major axis is parallel to the y-axis, and the lengths are given by \(2a\) and \(2b\). Here, \(a = \sqrt{9} = 3\) and \(b = \sqrt{4} = 2\). Therefore, the length of the major axis is \(2 \times 3 = 6\) and the minor axis is \(2 \times 2 = 4\).

Step 4: Identify the End Points of the Major and Minor Axes

The endpoints of the major axis (vertical) are at \((-5, 7+3) = (-5, 10)\) and \((-5, 7-3) = (-5, 4)\). The endpoints of the minor axis (horizontal) are at \((-5+2, 7) = (-3, 7)\) and \((-5-2, 7) = (-7, 7)\).

Step 5: Determine the Foci of the Ellipse

The distance from the center to the foci is given by \(c = \sqrt{a^{2} - b^{2}} = \sqrt{9 - 4} = \sqrt{5}\). The foci are situated along the major axis at \((-5, 7 + \sqrt{5})\) and \((-5, 7 - \sqrt{5})\).

Key Concepts

Standard Form of Ellipse

The standard form of an ellipse is an equation that helps us understand its size, shape, and orientation. It fits into the format: \( \frac{(x-h)^{2}}{b^{2}} + \frac{(y-k)^{2}}{a^{2}} = 1 \) for a vertical ellipse. Here,

• \((h, k)\) represents the center of the ellipse.
• \(a^2\) and \(b^2\) are the squares of the lengths of the semi-major and semi-minor axes, respectively.

In the given equation \( \frac{(x+5)^{2}}{4} + \frac{(y-7)^{2}}{9} = 1 \), it's clear that \(a^2 = 9\) and \(b^2 = 4\). Since \(a^2\) is greater than \(b^2\), this forms a vertical ellipse with the major axis aligned along the y-axis.
This standard form allows for quick identification of the ellipse's main components, facilitating the calculation of axes, the center, and the foci.

Axes of Ellipse

Ellipses have two principal axes, known as the major and minor axes.

• The **major axis** is the longer one, which in vertical ellipses is parallel to the y-axis.
• The **minor axis** is the shorter one, aligned parallel to the x-axis.

The lengths of these axes are determined by:

• The length of the major axis is \(2a\), where \(a = \sqrt{9} = 3\). Thus, its total length is \(6\).
• The length of the minor axis is \(2b\), where \(b = \sqrt{4} = 2\). Therefore, it measures \(4\).

Knowing these axes is crucial as they define the ellipse's shape and where the end points are located. For a vertical ellipse like ours, the endpoints of the major axis are found at the coordinates \((-5, 10)\) and \((-5, 4)\), while the minor axis endpoints are at \((-3, 7)\) and \((-7, 7)\).

Center of Ellipse

Identifying the center of an ellipse is essential as it serves as the reference point for plotting the rest of the ellipse. In standard form, the center is represented by the coordinates \((h, k)\). In our example,

• We extract the values from "+5" and "-7" in the equation, indicating our center is \((-5, 7)\).

The center is obtained from adjusting the equation terms:

• \((x+5)^2\) means shifting the ellipse 5 units to the left.
• \((y-7)^2\) means moving it 7 units upwards.

Understanding how to extract and interpret these shifts helps determine the position of the ellipse in the coordinate plane.

Foci of Ellipse

The foci are two special points inside an ellipse. They are crucial for understanding the ellipse's geometric properties. For ellipses, the sum of the distances from any point on the ellipse to the two foci is constant.
To find the foci:

• First, calculate the distance from the center to each focus using \(c = \sqrt{a^2 - b^2}\).
• In the given problem, this results in \(c = \sqrt{9 - 4} = \sqrt{5}\).

Since this is a vertical ellipse, the foci are along the major axis:

• The foci are positioned at \((-5, 7 + \sqrt{5})\) and \((-5, 7 - \sqrt{5})\).

These points help define how 'stretched' the ellipse is, influencing its overall shape. Knowing how to calculate and locate the foci is key in fully understanding the structure of an ellipse.