Problem 33 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length \(10 ;\) length of minor axis \(=4 ;\) center: \((-2,3)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse with the given conditions is \[((x + 2)^2) / 4 + ((y - 3)^2) / 25 = 1\].

Step 1: Identify the centre, semi-major axis and semi-minor axis

Given in the exercise, the center of the ellipse (\(h, k\)) is (-2, 3). The length of the major axis is given to be 10, therefore, the length of semi-major axis \(A\) is \(10/2 = 5\). The length of the minor axis is given to be 4, so the length of semi-minor axis \(B\) is \(4/2 = 2\).

Step 2: Substitute values into the standard equation

Substitute \(h = -2\), \(k = 3\), \(A = 5\), and \(B = 2\) into the standard equation to get \[((x + 2)^2) / 2^2 + ((y - 3)^2) / 5^2 = 1\].

Key Concepts

Major Axis of an Ellipse

The major axis of an ellipse is the longest diameter that passes through the center of the ellipse and connects the two furthest points. For a vertical ellipse like the one described in the exercise, the major axis is vertical. The length of the major axis is denoted as the distance between these two furthest points. Understanding the length of the major axis is crucial because it helps determine the semi-major axis which is simply half of the major axis. In our example, the major axis has a length of 10 units.

Minor Axis of an Ellipse

Conversely, the minor axis is the shortest diameter that passes through the center and connects the two closest points on the ellipse, and it is perpendicular to the major axis. For horizontal ellipses, the minor axis is horizontal, but in our problem, since the major axis is vertical, the minor axis is horizontal. The length of the minor axis provides valuable information in defining the overall shape of the ellipse. In the provided exercise, the minor axis has a length of 4 units.

Ellipse Center Coordinates

The center coordinates of an ellipse, represented as \(h, k\), are the midpoint of both the major and minor axes. The center dictates the position of the ellipse within the coordinate system. For the given exercise, the coordinates of the center are \(h, k\) = (-2, 3), which indicates the location from where the major and minor axes extend. Knowing the center is essential for constructing the standard form equation of the ellipse.

Semi-Major Axis

The semi-major axis of an ellipse is half the length of the major axis and extends from the center to one of the furthest points along the major axis. It is a vital component in the equation of an ellipse as it affects the curve's stretch. In our exercise, the semi-major axis (A) is half of 10, which results in a length of 5 units. The value of the semi-major axis is used to determine the denominator of the \(y\) term in the standard equation of an ellipse.

Semi-Minor Axis

Similarly, the semi-minor axis is half the length of the minor axis and connects the center to one of the closest points along the minor axis. The length of the semi-minor axis (B) influences how flat or elongated the ellipse appears. In the context of the discussed problem, the semi-minor axis is 4 units divided by two, giving us 2 units. This value is used in the denominator of the \(x\) term in the standard form of the ellipse's equation. With a semi-major axis of 5 and a semi-minor axis of 2, the complete standard equation of the given ellipse would be \(((x + 2)^2) / 2^2 + ((y - 3)^2) / 5^2 = 1\).