Problem 47 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed a conic in the form \(r=\frac{2 p}{1-e \cos \theta}\) that was symmetric with respect to the \(y\) -axis.

Step-by-Step Solution

Answer

Yes, the statement makes sense. The graph of the conic section in polar form \(r=\frac{2 p}{1-e \cos \theta}\) is indeed symmetric with respect to the y-axis.

Step 1: Writing the Condition for Symmetry

The condition to verify whether a function in the polar form is symmetric with respect to the y-axis is based on replacing \(\theta\) with \(-\theta\) and checking whether the functional form remains the same or not. Hence, we'll plug \(-\theta\) in place of \(\theta\) in the given equation \(r=\frac{2 p}{1-e \cos \theta}\) and observe the outcome.

Step 2: Substituting \(-\theta\) into the equation.

Replacing \(\theta\) by \(-\theta\) in our equation we get \(r=\frac{2 p}{1-e \cos(-\theta)}\). Since cosine function is an even function, thus \(\cos(\theta) = \cos(-\theta)\). Hence the expression will remain unaltered i.e., \(r=\frac{2 p}{1-e \cos \theta}\).

Step 3: Conclusion

Since the equation \(r=\frac{2 p}{1-e \cos \theta}\) remains the same even after replacing \(\theta\) with \(-\theta\), it is confirmed that our graph is symmetric about the y-axis.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to locate points on a plane by using a combination of radius (r) and angle (θ). Unlike the Cartesian system, where points are located using x and y coordinates, polar coordinates provide a different perspective.
In essence, the system is centered around the origin, also known as the pole. The angle, θ, is measured from the positive x-axis, and the radius, r, is the distance of the point from the origin.

• If r is positive, the point is located on the angle's direction.
• If r is negative, it appears in the opposite direction of θ.

Polar coordinates are exceptionally useful when dealing with curves like spirals and circles, as they can simplify equations and calculations.

Symmetry in Polar Equations

Symmetry plays a crucial role in understanding the behavior of graphs in polar equations. To determine symmetry, we often use specific substitutions. For example, checking symmetry with respect to the y-axis involves replacing θ with -θ in the equation. If the equation remains unchanged, it has y-axis symmetry.
Other types of symmetry in polar graphs include:

• Polar Axis Symmetry: Replace θ with π - θ.
• Origin Symmetry: Replace r with -r or θ with θ + π.

For conics expressed in polar form, this symmetry testing helps identify the nature and orientation of the graph, providing insights into its geometric properties.

Even Functions

In mathematics, an even function is one that satisfies the condition f(x) = f(-x) for all x in its domain. This property of even functions is particularly significant when graphing in polar coordinates.
The cosine function, for instance, is even because cos(θ) = cos(-θ). Thus, when substituting θ with -θ in our equation, the cosine term remains unchanged, aiding in identifying symmetry within the graph.
Understanding even functions can simplify problems involving symmetry, allowing for easier prediction and verification of graph behavior in various applications.

Graphing Conics

Graphing conic sections using polar coordinates can sometimes reveal properties more clearly than Cartesian coordinates. Conic sections include circles, ellipses, parabolas, and hyperbolas, each with unique equations and characteristics.
In polar coordinates, a common form is \[ r = \frac{ep}{1-e\cos\theta} \]where e represents the eccentricity and dictates the shape of the conic:

• Circle: e=0
• Ellipse: 0 < e < 1
• Parabola: e = 1
• Hyperbola: e > 1

Graphing these can unveil new insights about their symmetry and orientation, aiding in a deeper understanding of these geometric figures.