Problem 19 - Algebra 2

Question

Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. Then graph the ellipse. $$ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 $$

Step-by-Step Solution

Answer

Center: (0, 0); Foci: (±4, 0); Major axis: 10; Minor axis: 6.

Step 1: Identify the Form of the Equation

The given equation is \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \). This is in the standard form of an ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a > b \).

Step 2: Determine the Values of a and b

In the equation \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), we identify \( a^2 = 25 \) and \( b^2 = 9 \). Solving for \( a \) and \( b \), we find \( a = \sqrt{25} = 5 \) and \( b = \sqrt{9} = 3 \).

Step 3: Identify the Center of the Ellipse

Since the equation has the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) where \( h \) and \( k \) are zero, the center of the ellipse is at the origin \((0, 0)\).

Step 4: Calculate the Lengths of the Major and Minor Axes

The length of the major axis is \(2a = 2 \times 5 = 10\) and the length of the minor axis is \(2b = 2 \times 3 = 6\).

Step 5: Find the Coordinates of the Foci

For an ellipse, the distance \(c\) from the center to each focus is defined by \(c^2 = a^2 - b^2\). Calculating gives \(c^2 = 25 - 9 = 16\), so \(c = 4\). The foci are at \((\pm c, 0)\) which are \((\pm 4, 0)\).

Key Concepts

Center of Ellipse

The center of an ellipse is a crucial part of understanding its structure. It serves as the starting point from which the entire shape radiates. For any ellipse described by the equation \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the center is given by the coordinates \((h, k)\). In our specific example, the equation \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) has no added term with \(h\) or \(k\), meaning both are zero.

Consequently, the center of this ellipse is positioned at the origin, \((0, 0)\). This location is where the major and minor axes intersect and is symmetrical along these axes. Understanding the place of the center is essential as it forms the basis for plotting the rest of the ellipse as the axes and foci derive from it.

Major and Minor Axes

The major axis of an ellipse is the longest diameter that passes through the center. In our exercise, the standard form equation \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) guides us to observe that \(a^2 = 25\) and \(b^2 = 9\). Here, \(a = 5\) and \(b = 3\).

Since \(a > b\), the major axis is aligned with the x-axis. Its length, calculated as \(2a = 2 \times 5 = 10\), spans horizontally. The minor axis, being the shorter dimension, aligns with the y-axis with a length \(2b = 2 \times 3 = 6\).

These axes are perpendicular and intersect at the center of the ellipse.

• Major axis: spans 10 units along x-coordinates from \(-5\) to \(5\)
• Minor axis: spans 6 units along y-coordinates from \(-3\) to \(3\)

Remember, the major axis is always longer than the minor axis and defines the direction in which the ellipse is elongated.

Foci of Ellipse

The foci (singular: focus) are two critical points inside an ellipse. They play a vital role in its mathematical definition. The distance between a point on the ellipse and each focus is constant when summed. To determine the foci's positions, we calculate \(c\) using \(c^2 = a^2 - b^2\).

In our example:

• \(a^2 = 25\)
• \(b^2 = 9\)
• \(c^2 = 25 - 9 = 16\)
• \(c = \sqrt{16} = 4\)

Since the major axis is horizontal, the foci lie on the x-axis. Therefore, their coordinates are \((-4, 0)\) and \((4, 0)\).

The location of the foci helps to understand the shape's eccentricity and how much it deviates from being a perfect circle. The larger the distance between the foci relative to the major axis’s length, the more ‘stretched’ the ellipse looks.