Problem 54 - Algebra 2

Question

Which equation represents a line of symmetry for the ellipse with equation \(x^{2}+9 y^{2}=9 ?\) \(\begin{array}{lllll}{\text { A. } y=} & {-x} & {\text { B. } y=0} & {\text { C. } y=x} & {\text { D. } x y=1}\end{array}\)

Step-by-Step Solution

Answer

The equations that represent lines of symmetry for the circle are: \(y = x\) (Option C), \(y = 0\) (Option B), and \(y = -x\) (Option A).

Step 1: Analyzing the given equation

The given equation is \(x^2 + 9y^2 = 9\). Dividing by 9, it can be written in the form \(x^2/a^2 + y^2/b^2 = 1\), where \(a = b = 3\), which denotes a circle with radius 3, centered at the origin.

Step 2: Identifying lines of symmetry

As it is a circle, every line passing through the origin is a line of symmetry. This means that the following equations represent lines of symmetry for this circle: \(y = x\), \(y = -x\), \(y = 0\) and \(x = 0\).

Step 3: Choosing the correct answer

Given the answer choices, the lines of symmetry for this circle that we found are represented by options A, B, and C.

Key Concepts

Ellipse Equations

Understanding ellipses begins with the equation that defines them. An ellipse is typically given by the formula \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where:

• \(a\) is the semi-major axis
• \(b\) is the semi-minor axis
• When \(a = b\), the ellipse becomes a circle.

In the context of this particular exercise, the equation \(x^2 + 9y^2 = 9\) is rewritable as \(\frac{x^2}{3^2} + \frac{y^2}{1^2} = 1\). This indicates that \(a = 3\) and \(b = 1\). In fact, from simplification, \(a\) and \(b\) could both be taken as \(3\), making it a special case of a circle. Understanding these fundamentals is essential, as they frame how we analyze and interpret ellipses in coordinate geometry.

Lines of Symmetry

Symmetry lines of an ellipse (or circle) are pivotal in identifying its properties and behavior in coordinate geometry. In general, any line through the center of an ellipse can serve as a line of symmetry. For circles, like in this example, any diametric line is a line of symmetry because a circle is perfectly uniform from its center in every direction.
In the problem at hand, after slicing through mathematical operations, it’s revealed that several options are lines of symmetry:

• \(y = x\) - a diagonal line through the origin, dividing the circle into two symmetrical halves.
• \(y = -x\) - another diagonal line with the same property but slanted in the opposite direction.
• \(y = 0\) - the x-axis, effectively bisecting the circle horizontally.

These specific lines of symmetry confer to the solution of the exercise, enhancing our understanding of predictable geometric symmetry properties.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, facilitates the understanding of geometrical figures through algebraic equations and coordinates.This branch of mathematics unifies geometry and algebra by expressing shapes in an algebraic form. In the study of ellipses or circles, it's fruitfully applied by breaking down their equations and interpretations.
Through the exercise, traversing from the equation \(x^2 + 9y^2 = 9\) to understanding the symmetry via coordinate axes is a good example of this application. Coordinate geometry allows the visualization of not only the geometry of the shape but also its spatial relationships and properties. It's this approach that helps identify varying solutions, such as symmetry lines that accurately reflect the circle's or ellipse’s nature being studied.
Applying these principles aids students in grasping the better the intrinsic properties of shapes, thus not only solving the problem at stake but forming strong foundational geometrical insights.