Problem 50 - Algebra and Trigonometry

Question

Identify the conic and graph the equation: $$ r=\frac{4 \sec \theta}{2 \sec \theta-1} $$

Step-by-Step Solution

Answer

The equation represents a circle with center at (2,0) and radius 2.

Step 1: Rewrite the equation in Cartesian coordinates

To start with, observe that given equation can be manipulated into the form \(r=\frac{a}{1±e\cos \theta}\) or \(r = \frac{a}{1±e\sin \theta}\), which would indicate a conic section in polar coordinates. Next, convert from polar to Cartesian coordinates by using the relationships: \(x = r\cos\theta\) and \(y = r\sin\theta\). By replacing \(r\) with \(\frac{x}{\cos\theta}\) from the equation \(x = r\cos\theta\), the given equation becomes: \(\frac{x}{\cos\theta} = \frac{4\sec\theta}{2\sec\theta - 1}\).

Step 2: Simplify the equation

Simplify both sides of the equation. On the left side, \(\cos\theta\) and \(\sec\theta\) cancel each other being reciprocals. The equation simplifies to: \(x = \frac{4}{2 - \cos\theta}\). Now, replace \(\cos\theta\) with \(\frac{x}{r}\) to get: \(x = \frac{4r}{2r - x}\). On simplifying we get: \(x^2 - 4r^2 + 4x = 0\). Replace \(r^2\) with \(x^2 + y^2\). This gives the equation: \(x^2 + y^2 - 4x = 0\). Complete the square by adding 4 to both sides, resulting in: \(x^2 - 4x + 4 + y^2 = 4\). Simplify the equation to: \((x-2)^2 + y^2 = 4\).

Step 3: Identify the conic

The resulting equation is in the standard form of a circle. It represents a circle with center (2,0) and radius 2.

Step 4: Graph the Conic

Now, graph the conic. Sketch a circle with center at (2,0) and radius 2 in the Cartesian plane

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points in a plane. Instead of using the standard Cartesian coordinate system with x and y axes, polar coordinates describe a point based on its distance from a reference point, known as the pole (similar to an origin), and the angle from a reference direction, usually the positive x-axis. These coordinates are expressed in terms of radius \( r \) and angle \( \theta \). Points in polar coordinates are denoted as \( (r, \theta) \), where \( r \) is the distance from the pole, and \( \theta \) is the angle from the reference direction.

Using polar coordinates is particularly useful when working with conic sections. This is because the equations can often be simpler in form, leading to easier manipulation when identifying the type of conic. In the given exercise, the equation \( r=\frac{4 \sec \theta}{2 \sec \theta-1} \) is provided in polar form, which helps recognize potential conic shapes.

• The key is to reformat the equation to identify whether it’s a circle, ellipse, parabola, or hyperbola.
• The expression often involves trigonometric functions like \( \sec \theta \), linking polar angles to radius.

Cartesian Coordinates

Cartesian coordinates are the most commonly used coordinate system, defined by x and y axes. Each point in the plane is specified by an ordered pair \( (x, y) \). To convert from polar coordinates to Cartesian coordinates, the following relationships are used:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These relationships allow us to describe points in the familiar grid-like layout of the Cartesian plane.

In this exercise, the task was to convert the given polar equation into Cartesian form. The conversion process involved substituting the polar terms \( r \cos \theta \) and \( r \sin \theta \) with \( x \) and \( y \), respectively. This leads to a new equation in terms of \( x \) and \( y \), making it easier to classify the conic section. The rewritten equation, after conversion and simplification, becomes \( (x-2)^2 + y^2 = 4 \), fitting the standard form of a circle.

Circle Equation

The circle is one of the fundamental shapes in geometry and can be easily represented in both coordinate systems. In Cartesian coordinates, a circle with center \( (h, k) \) and radius \( r \) is expressed with the equation \((x-h)^2 + (y-k)^2 = r^2\).

In this example, through the step-by-step simplification and conversion from the polar coordinate system, the exercise yielded the equation \( (x-2)^2 + y^2 = 4 \). Here:

• The center of the circle is \( (2, 0) \)
• The radius of the circle is \( 2 \), because \( r^2 = 4 \) gives \( r = \sqrt{4} = 2 \)

Recognizing the equation's form helps identify the conic section, confirming it as a circle based on both its center and radius. Simplifying and transforming equations using standard forms is crucial in graph identification and classification.

Graphing Conics

Graphing conic sections involves sketching the shapes described by their respective equations. They include circles, ellipses, parabolas, and hyperbolas. Each can be identified and drawn following their distinct characteristics. For circles:

• Determine the center point from the equation.
• Identify the radius.
• Sketch a round shape that extends equally in all directions from the center, matching the radius.

The equation from the exercise, \( (x-2)^2 + y^2 = 4 \), describes a circle with a center at \( (2, 0) \) and a radius of 2.

To graph this:

• Locate the center point (2, 0) on the Cartesian plane.
• From the center, measure 2 units in all directions (up, down, left, and right) to outline the circle's boundary.
• Draw the circle, ensuring it's symmetric about its center.

Creating accurate graphs of conic sections visually reinforces understanding and provides insights into their geometric properties.