Problem 7 - Intermediate Algebra

Question

Write the standard form of the equation of an ellipse centered at the origin and symmetric to both axes.

Step-by-Step Solution

Answer

The standard form is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Step 1: Understand the Standard Form of Ellipse

The standard form of the equation of an ellipse centered at the origin is given by \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively. In this form, \(a\) and \(b\) directly influence the shape and orientation.

Step 2: Recognize the Symmetry Condition

Since the ellipse is symmetric to both axes and centered at the origin, we actually do not need to further modify the form from that given in Step 1. This symmetry condition ensures that if \((x, y)\) is on the ellipse, then \((-x, y)\), \((x, -y)\), and \((-x, -y)\) will also be on it.

Step 3: Consider Special Symmetric Cases (if needed)

Since the problem does not specify if \(a\) is greater than \(b\) or vice versa, we consider both cases without needing specific values: - If \(a = b\), the ellipse is a circle, and the equation reduces to \(x^2 + y^2 = r^2\) where \(r\) is the radius.- If \(a eq b\), it stays an ellipse as described in Step 1.

Key Concepts

Standard Form of Ellipse

The standard form of the ellipse equation is essential for understanding the characteristics of ellipses. This form is typically expressed as: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Here, the ellipse is centered at the origin \((0, 0)\), with \(a\) representing the semi-major axis, and \(b\) representing the semi-minor axis. This equation allows you to see at a glance the orientation and relative size of the ellipse's axes. The denominator under each square term indicates how the ellipse spreads out along the x or y-axis. Specifically:

• If \(a > b\), the ellipse is wider than it is tall, stretching more along the x-axis.
• If \(b > a\), the ellipse is taller than it is wide, stretching more along the y-axis.

Understanding this form can dramatically aid in sketching and recognizing the orientation of an ellipse simply from its equation.

Symmetry in Ellipses

An essential feature of ellipses, especially those centered at the origin, is symmetry. An ellipse symmetric about the x-axis and y-axis implies that any point \((x, y)\) on the ellipse will have symmetric counterparts at the other quadrants:

• \((-x, y)\)
• \((x, -y)\)
• \((-x, -y)\)

This symmetry simplifies the understanding of ellipses in many ways. For graphing, it means once you know one segment, you automatically know the rest, significantly reducing the complexity of the task. The equation \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] inherently accounts for this symmetry and requires no modifications to exhibit these characteristics when centered at the origin.

Semi-Major and Semi-Minor Axes

The semi-major and semi-minor axes are critical in defining the shape and size of an ellipse. The terms are used to describe the longest and shortest radii of an ellipse, respectively.

• Semi-Major Axis (\(a\)): This is the longest radius, which extends from the center to the farthest ellipse edge. If \(a = b\), the shape is a circle.
• Semi-Minor Axis (\(b\)): This is the shortest radius, extending to the closest edge of the ellipse from the center.

If \(a > b\), the ellipse is horizontally oriented, meaning it stretches further along the x-axis. Conversely, if \(b > a\), it is vertically oriented, indicating it stretches more along the y-axis. These axes are central to understanding the geometric properties of an ellipse and assist in various applications, from architectural design to planetary orbits in astronomy.