Problem 33 - A Graphical Approach to College Algebra

Question

Find an equation for each ellipse. Major axis of length 6 ; foci \((0,2)\) and \((0,-2)\)

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{5} + \frac{y^2}{9} = 1\).

Step 1: Understand the Components of the Ellipse

An ellipse has a center, vertices along the major axis, and foci. The equation form is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) where \((h,k)\) is the center, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. The distance from the center to each focus is \(c\).

Step 2: Locate the Center of the Ellipse

Since the foci are \((0,2)\) and \((0,-2)\), they are symmetrically placed about the x-axis, so the center of the ellipse is \((0,0)\).

Step 3: Determine the Semi-Major Axis

The length of the major axis is 6, so the semi-major axis \(a\) is half of that: \(a = \frac{6}{2} = 3\).

Step 4: Calculate the Distance to the Foci

The distance from the center \((0,0)\) to the foci \((0,2)\) or \((0,-2)\) is 2. Thus, \(c = 2\).

Step 5: Use the Relationship Between a, b, and c

For an ellipse, the relationship \(c^2 = a^2 - b^2\) holds. Here, \(c = 2\) and \(a = 3\), we find \(b\) by:\[c^2 = a^2 - b^2 \rightarrow 2^2 = 3^2 - b^2 \rightarrow 4 = 9 - b^2 \quad \therefore b^2 = 5\].

Step 6: Write the Equation of the Ellipse

Since the center is \((0,0)\), the equation simplifies to:\[\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\]\[\frac{x^2}{5} + \frac{y^2}{9} = 1\].

Key Concepts

Major Axis

The major axis of an ellipse is a crucial component, as it is the longest diameter of the ellipse and determines the width or height of the shape, depending on its orientation. It passes through the center of the ellipse and both foci.
The major axis is always the greater value when compared to the minor axis. Its length is formally defined as the distance between the two vertices at either end of the axis. In the problem, the major axis is given as 6 units long.

• This means each half of the major axis, known as the semi-major axis, is 3 units long.
• The major axis is oriented along the y-axis here since the foci are given as (0,2) and (0,-2).

This is an important feature because it influences how we place and determine the other components of the ellipse.

Foci

The foci of an ellipse are two fixed points located along its major axis. They are instrumental in defining the shape of an ellipse, as any point on the ellipse is equidistant from the sum of the distances to each focus.
In our example, the foci are located at (0,2) and (0,-2). This tells us that the ellipse is vertically oriented because the foci lie along the y-axis.

• These points are each 2 units away from the center of the ellipse.
• The notation for the distance from the center to a focus is usually denoted as "c." In this case, c = 2.

Understanding the location of the foci helps us not only position the ellipse in coordinate geometry but also identify its overall orientation.

Center of Ellipse

The center of an ellipse is the midpoint between the two foci. It is a pivotal point from which we measure the axes.
Since the foci are symmetric about the x-axis in this problem, the center can be directly calculated as (0,0).

• This point serves as the origin of the ellipse's coordinate system.
• The coordinates of the center are traditionally represented as (h,k), where in this example, both h and k equal zero.

The position of the center relative to other ellipse components helps us simplify the equation form.

Semi-Major Axis

The semi-major axis is half the length of the major axis. It directly extends from the center to one end of the ellipse, where you hit a vertex.
In our calculation, the major axis is 6 units in total, hence the semi-major axis is given by: a = \(\frac{6}{2}\) = 3.

• Its length determines the stretch of the ellipse along the major axis.
• The semi-major axis is greater than the semi-minor axis, confirming again the extent to which the ellipse is elongated.

The distance "a" is also prominently featured in the standard ellipse equation: \(a^2\) appears as the denominator in the equation, representing this semi-axis's squared value.

Semi-Minor Axis

The semi-minor axis is the shorter radius of the ellipse, perpendicular to the semi-major axis. It extends from the center to the edge of the ellipse, halfway along the minor axis.
In this example, to find the semi-minor axis, we use the relationship between the axes and foci: \(c^2 = a^2 - b^2\).

• With c = 2 and a = 3, we have 4 = 9 - b^2, solving for b^2 gives b^2 = 5.
• The semi-minor axis b = \(\sqrt{5}\).

The semi-minor axis complements the semi-major axis and together they provide a full picture of the ellipse's dimensions. It also serves as another component in the standard form ellipse equation.