Problem 13 - 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. $$ x^{2}+9 y^{2}=1 $$

Step-by-Step Solution

Answer

The ellipse is \(\frac{x^2}{1} + \frac{y^2}{(1/3)^2} = 1\). Major axis: (1, 0), (-1, 0). Minor axis: (0, 1/3), (0, -1/3). Foci: \(\left(\pm \frac{2\sqrt{2}}{3}, 0\right)\).

Step 1: Identify the General Form of the Ellipse Equation

The general form of an ellipse equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) for ellipses aligned with the coordinate axes. Identify the coefficients in the given equation \( x^2 + 9y^2 = 1 \). In this equation, recognize that it is already in a format similar to the general form, with \( a^2 = 1 \) and \( b^2 = \frac{1}{9} \).

Step 2: Convert Equation to Standard Form

From identifying \( a^2 = 1 \) and \( b^2 = \frac{1}{9} \), rewrite the equation in the format \( \frac{x^2}{1} + \frac{y^2}{\left(\frac{1}{3}\right)^2} = 1 \). This confirms that the given equation is \( \frac{x^2}{1^2} + \frac{y^2}{\left(\frac{1}{3}\right)^2} = 1 \), its standard form specifically for this ellipse.

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

Compare \( a^2 \) and \( b^2 \): since \( a^2 > b^2 \), the major axis is along the x-axis. The lengths of the semi-major and semi-minor axes are \( a = 1 \) and \( b = \frac{1}{3} \) respectively.

Step 4: Identify the Endpoints of the Axes

The endpoints of the major axis are at \((\pm a, 0) = (\pm 1, 0)\), and the endpoints of the minor axis are at \((0, \pm b) = \left(0, \pm \frac{1}{3}\right)\). This gives us the points (1, 0), (-1, 0) for the major axis and (0, 1/3), (0, -1/3) for the minor axis.

Step 5: Calculate the Foci of the Ellipse

For an ellipse, the formula for the distance to the foci along the major axis is \( c^2 = a^2 - b^2 \). Calculate \( c \) as follows: \( c^2 = 1^2 - \left(\frac{1}{3}\right)^2 \). This simplifies to \( c^2 = 1 - \frac{1}{9} = \frac{8}{9} \), hence \( c = \frac{\sqrt{8}}{3} \). The foci are at \((\pm c, 0) = \left(\pm \frac{2\sqrt{2}}{3}, 0 \right)\).

Key Concepts

Standard Form of Ellipse

The standard form of an ellipse equation is crucial for understanding and solving problems related to ellipses. An ellipse's standard form aligns with the coordinate axes and is typically expressed as:

• Horizontal Ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
• Vertical Ellipse: \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \)

This form provides valuable information about the ellipse, such as its orientation and the lengths of its axes. For instance, the equation \( x^2 + 9y^2 = 1 \) becomes \( \frac{x^2}{1} + \frac{y^2}{(1/3)^2} = 1 \) when rewritten in standard form. Here, the coefficients correspond to the squares of the semi-axis lengths, with \( a = 1 \) and \( b = \frac{1}{3} \). This means we have a horizontal ellipse, as \( a > b \). Understanding this standard form is a stepping stone to further analyzing the ellipse's properties, such as its axes and foci.

Major and Minor Axes of Ellipse

The major and minor axes of an ellipse are the longest and shortest diameters running through its center. Identifying these axes helps to understand the basic shape and size of the ellipse.
For our ellipse with equation \( \frac{x^2}{1} + \frac{y^2}{(1/3)^2} = 1 \), we have:

• Major Axis: Along the x-axis, because \( a^2 = 1 \) is greater than \( b^2 = \frac{1}{9} \).
• Semi-Major Axis Length: \( a = 1 \).
• Minor Axis: Along the y-axis.
• Semi-Minor Axis Length: \( b = \frac{1}{3} \).

The endpoints of these axes are crucial for depicting the boundaries of the ellipse:

• Major Axis Endpoints: \( (1, 0) \) and \( (-1, 0) \).
• Minor Axis Endpoints: \( \left(0, \frac{1}{3}\right) \) and \( \left(0, -\frac{1}{3}\right) \).

These axes provide the backbone onto which the ellipse is oriented, dictating the span of the ellipse both horizontally and vertically.

Foci of Ellipse

The foci of an ellipse are two central points that play a crucial role in defining its shape. Located along the major axis, these points help us understand the eccentricity of the ellipse.
For our present ellipse example, the foci can be calculated using the equation:

• \( c^2 = a^2 - b^2 \)

Substitute the known values: \( c^2 = 1^2 - \left(\frac{1}{3}\right)^2 = \frac{8}{9} \). Thus, \( c = \frac{\sqrt{8}}{3} \), which simplifies to about \( \frac{2\sqrt{2}}{3} \).
The foci for the ellipse are then positioned at:

• \( \left(\pm \frac{2\sqrt{2}}{3}, 0\right) \)

Placing the foci along the major axis further cements the geometry of the ellipse and they are always inside the ellipse, precisely along its longest stretch. These focal points are significant as they help maintain the balance of the ellipsoid shape, drawing the ellipse around them.