Problem 36 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$(x-3)^{2}+(y+4)^{2}=25$$

Step-by-Step Solution

Answer

The polar equation is \( r = 10\cos\theta + 10\sin\theta \).

Step 1: Identify the Cartesian Circle Equation

The given equation \((x-3)^2 + (y+4)^2 = 25\) is the equation of a circle in Cartesian coordinates. The equation is of the form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine the Center and Radius

From the equation \((x-3)^2 + (y+4)^2 = 25\), we can identify that the center of the circle is \((h, k) = (3, -4)\) and the radius \(r\) is \(\sqrt{25} = 5\).

Step 3: Convert Center to Polar Coordinates

In polar coordinates, a point \((x, y)\) is expressed as \((r, \theta)\). Calculate the polar coordinates of the center \((3, -4)\). The radius \(r_c = \sqrt{3^2 + (-4)^2} = 5\) and angle \(\theta\) is \(\tan^{-1}\left(\frac{-4}{3}\right)\).

Step 4: Translate Circle Equation to Polar Form

In polar form, a circle centered at a point \((r_c, \theta_c)\) with radius \(r\) is expressed as: \[ r'(\theta) = r_c \cos(\theta - \theta_c) + \sqrt{r^2 - (r_c \sin(\theta - \theta_c))^2} \]. Apply this using the calculated center and radius. Here, we need to simplify: \(\theta_c = \tan^{-1}\left(\frac{-4}{3}\right)\).

Step 5: Simplified Equation in Polar Coordinates

Since the center lies at \((3, -4)\) with radius 5 in polar coordinates, using the substitution \((x, y) = (r\cos(\theta), r\sin(\theta))\), the polar equation becomes: \[ r = 10\cos\theta + 10\sin\theta \]. This represents the same circle in terms of polar coordinates.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a system used to uniquely specify points in a plane using ordered pairs of numbers. The system is based on two perpendicular lines, known as the x-axis and the y-axis. Each point in this system is defined by two numbers:

• The x-coordinate: The horizontal distance from the origin.
• The y-coordinate: The vertical distance from the origin.

For example, in the Cartesian plane, a point might be written as (3, -4), indicating that you move 3 units to the right along the x-axis and 4 units down along the y-axis from the origin (0,0).

Understanding Cartesian coordinates helps in defining the positions of points geometrically and is the foundation for writing equations that represent geometric shapes, such as lines and circles.

Circle Equation

The circle equation in Cartesian coordinates is expressed as \((x-h)^2 + (y-k)^2 = r^2\), where:

• \((h, k)\) are the coordinates of the center of the circle.
• \(r\) is the radius of the circle.

For instance, given the equation \((x-3)^2 + (y+4)^2 = 25\), we can identify the circle's center as (3, -4) and the radius as 5 because \(\sqrt{25} = 5\).

This equation tells us that any point \((x, y)\) on the circle is exactly 5 units away from the center at (3, -4). Recognizing the standard circle equation is essential for converting it to other forms, like polar coordinates, for various applications in mathematics and physics.

Radius

The radius of a circle is a crucial concept in geometry. It is defined as the distance from the center of the circle to any point on its circumference. The radius is constant for a given circle, making it a defining feature of that circle.

For example, based on the equation \((x-3)^2 + (y+4)^2 = 25\), the radius can be found by taking the square root of the constant term on the right. Hence, the radius \(r = \sqrt{25} = 5\).

• In practical terms, if you start at the center of a circle (3, -4), the radius tells you how far you need to move in any direction to reach the edge of the circle.

Understanding the radius is important not only for identifying the size of a circle but also for conversions to other coordinate systems.

Angle Conversion

Angle conversion is necessary when dealing with different coordinate systems like Cartesian and polar coordinates. In polar coordinates, angles are used to specify the direction of a radius from the center of the circle.

For example, if a point is given in Cartesian coordinates as (3, -4), converting to polar coordinates involves finding the angle \(\theta\) using the tangent function, \[\theta = \tan^{-1}\left(\frac{-4}{3}\right)\].

• These calculations help shift the focus from a static x and y approach to a dynamic radius and angle perspective.
• This angle indicates the inclination of the line connecting a point on the circle to its center, measured from the positive x-axis.
• Accurate angle conversion is key when interpreting geometrical positions in polar forms, especially for translating equations or modeling periodic phenomena.

Overall, angle conversion aids in navigating between coordinate systems that serve distinct purposes in calculations and visualizations.