Problem 32 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length \(12 ;\) length of minor axis \(=6 ;\) center: \((0,0)\)

Step-by-Step Solution

Answer

The standard form equation of the ellipse is \(\frac{x^2}{36} + \frac{y^2}{9} = 1\).

Step 1: Identify the given data

The major axis is horizontal with length \(12\) units, the minor axis has length \(6\) units and the center of the ellipse is at the origin \((0,0)\).

Step 2: Determine the values of \(a\) and \(b\)

An ellipse is defined by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where the center of the ellipse is at the origin, \(a\) represents the semi-major axis, and \(b\) is the semi-minor axis. Given that the lengths of the major and minor axes are \(12\) and \(6\) respectively, these lengths are twice the corresponding semi-axes. Therefore, the semi-major axis \(a = \frac{12}{2} = 6\) and the semi-minor axis \(b = \frac{6}{2} = 3\).

Step 3: Substitute the values of \(a\) and \(b\) into the equation

In this case, \(a = 6\) and \(b = 3\). Substituting into the general equation of ellipse gives us \(\frac{x^2}{6^2} + \frac{y^2}{3^2} = 1\), which simplifies to \(\frac{x^2}{36} + \frac{y^2}{9} = 1\).

Key Concepts

Major and Minor Axis of an Ellipse

Understanding the major and minor axes of an ellipse is crucial when exploring its equation in standard form. The major axis is the longest diameter of the ellipse and runs through its center, connecting the two furthest points of the ellipse. In a horizontal ellipse, this axis is along the horizontal plane. Conversely, the minor axis is the shortest diameter and also runs through the center, connecting the two closest points of the ellipse on the vertical plane if the ellipse is horizontal.
When given the lengths of the major and minor axes, like in our exercise example with lengths of 12 and 6 respectively, it's essential to know that these lengths actually represent double the lengths of the semi-major and semi-minor axes. Thus, to find the values for our equation, we divide these numbers by two. This concept is a common source of confusion for students, so remember: the lengths provided are the full diameters, not the radii. In simple terms, think of the major axis as the 'widest' part of the ellipse and the minor axis as the 'narrowest' part, with the center being the balance point.

Center of an Ellipse

The center of an ellipse is the point where its major and minor axes intersect. It is the internal balance point around which the ellipse is symmetrically shaped. In many problems, including ours, the center is at the origin, denoted as (0,0). The coordinates of the center are crucial in the standard form equation of the ellipse because they provide the 'h' and 'k' values in the equation \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), which represent the horizontal and vertical shifts from the origin.
However, when the center is at the origin as in our exercise, 'h' and 'k' are both 0, simplifying the equation. Students should take note that the center's coordinates will affect the overall equation of the ellipse. If the center is not at the origin, the shifts must be accounted for in the equation, altering the x and y values within the equation.

Semi-Major and Semi-Minor Axis

The terms 'semi-major' and 'semi-minor' axis refer to half the length of their respective major and minor axes. The semi-major axis, denoted by 'a', is half the distance across the ellipse at its widest point. The semi-minor axis, 'b', is half the distance across the ellipse at its narrowest point.
In the context of our textbook problem, the length of the major axis is 12, making the semi-major axis \(a = \frac{12}{2} = 6\). Similarly, with a minor axis of 6, the semi-minor axis is \(b = \frac{6}{2} = 3\). These values of 'a' and 'b' are then squared and placed as the denominators in the standard form equation of the ellipse. It's important for students to grasp this concept; the semi-major axis (a) always corresponds with the x-square term, and the semi-minor axis (b) with the y-square term in the equation, assuming the ellipse is horizontally oriented. If the ellipse were vertical, 'a' and 'b' would switch roles in relation to x and y.