Problem 5 - College Algebra

Question

What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the \(y\) -axis?

Step-by-Step Solution

Answer

The ellipse is symmetric with respect to both the x-axis and the y-axis.

Step 1: Identify the Standard Form of the Ellipse

The standard form of the equation of an ellipse centered at the origin with its foci along the y-axis is given by: \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \( a > b \).

Step 2: Determine the Axes of Symmetry

For an ellipse centered at the origin, its symmetry is aligned with both coordinate axes. Specifically, it is symmetric with respect to the x-axis and the y-axis, as any vertical or horizontal line through the center (origin) will create mirror images on either side of the ellipse.

Step 3: Verify Symmetry Along the x-Axis

Replace \( y \) with \( -y \) in the ellipse equation: \( \frac{x^2}{b^2} + \frac{(-y)^2}{a^2} = 1 \). Simplifying gives \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), confirming symmetry along the x-axis.

Step 4: Verify Symmetry Along the y-Axis

Replace \( x \) with \( -x \) in the ellipse equation: \( \frac{(-x)^2}{b^2} + \frac{y^2}{a^2} = 1 \). Simplifying gives \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), confirming symmetry along the y-axis.

Key Concepts

The Standard Form of an Ellipse

When you're examining an ellipse, understanding its equation in standard form is essential. This form gives insight into the ellipse's properties and orientation. The standard form of an ellipse centered at the origin, with its foci along the y-axis, is expressed as: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]where \( a > b \). The variables \( x \) and \( y \) represent the coordinates on a Cartesian plane. Meanwhile, \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively. If the foci are aligned along the y-axis, it means that the semi-major axis extends in the vertical direction, dictating that \( a \) is paired with the \( y \) term. This form of the equation helps you immediately identify how the ellipse is stretched or compressed and in which direction.

Axes of Symmetry

Ellipses are interesting shapes because they have specific lines of symmetry, imparting balance and aesthetic to their form. For an ellipse with its center at the origin, it exhibits symmetry across its axes.

• Vertical Symmetry: This type of symmetry exists when the ellipse's vertical line through the center mirrors its top and bottom halves.
• Horizontal Symmetry: The horizontal line through the center reflects the left and right of the ellipse, illustrating this symmetry.

By examining these properties of symmetry, you can imagine folding the ellipse along the x-axis or the y-axis. Both folds would result in the ellipse's halves matching up perfectly. This symmetry is a useful property when plotting or analyzing ellipses on a graph.

Coordinate Axes Symmetry

The symmetry of an ellipse relative to the coordinate axes is crucial for understanding its geometry and plotting. An ellipse centered at the origin is symmetric about both the x-axis and y-axis. These are its coordinate axes.To demonstrate symmetry:

• Replace \( y \) with \( -y \) in the equation: \( \frac{x^2}{b^2} + \frac{(-y)^2}{a^2} = 1 \). Simplifying, you return to \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), proving symmetry about the x-axis.
• Similarly, replace \( x \) with \( -x \): \( \frac{(-x)^2}{b^2} + \frac{y^2}{a^2} = 1 \). Simplification confirms the symmetry about the y-axis with \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \).

This symmetrical nature means that each side of the ellipse is an identical "mirror" image, simplifying many calculations and visualizations in mathematics. Symmetry along the axes implies if a point \((x,y)\) lies on the ellipse, the points \((x,-y)\), \((-x,y)\), and \((-x,-y)\) also lie on it.