Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the polar equation \(r=2(h \cos \theta+k \sin \theta)\) to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.

Step-by-Step Solution

Answer

The polar equation represents a circle with radius 2 and center at rectangular coordinates \((h, k)\). The rectangular form of the given polar equation verifies this.

Step 1: Convert Polar to Rectangular Form

Using the relationship between polar coordinates and rectangular coordinates, where \(x = r \cos \theta\) and \(y = r \sin \theta\), the given equation \(r=2(h \cos \theta+k \sin \theta)\) can be converted to rectangular form. This would yield \(x = 2h\) and \(y = 2k\).

Step 2: Verify the Equation Represents a Circle

In rectangular coordinates, the general equation of a circle is \((x-a)^2 + (y-b)^2 = r^2\), where \((a, b)\) is the center and \(r\) is the radius. Comparing this with our equation from step 1, a circle can be represented with center at \((h, k)\) and radius 2.

Step 3: Find the Radius and the Coordinates of the Center

From the equation of the circle derived in Step 2, it is evident that the radius of the circle is 2. Also, the center of the circle in rectangular coordinates is \((h, k)\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of the traditional

• horizontal "x" axis
• and vertical "y" axis found in rectangular coordinates

polar coordinates specify a point by:

• its distance from the origin, called the radius (\( r \)),
• and an angle (\( \theta \)) from a fixed direction, typically the positive x-axis.

This system is especially useful in scenarios related to circular motion and angles, as it directly measures both these quantities. For instance, radar and navigation systems commonly use polar coordinates. The polar equation given in the exercise refers to a scenario where the radius is expressed as a linear combination of sine and cosine functions.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe each point in the plane with an

• \( x \)
• and a \( y \) value.

Transformation from polar to rectangular coordinates uses simple trigonometry:

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

These equations make it easy to convert between the two coordinate systems. In the provided exercise, the polar equation was transformed into rectangular form using these formulas. This allows us to analyze and graph the equation more easily on a standard coordinate plane.

Circle Equation

The standard equation of a circle in a plane is y = (x-a)^2 + (y-b)^2 = r^2where:

• \((a, b)\) are the coordinates of the center
• \(r\) is the radius.

This equation confirms that all points (\( x, y \)) are at the same distance (radius) from the center. In our conversion from polar to rectangular coordinates, the transformed equation retains this circular form. Thus, it verifies that the original polar equation also represents a circle, maintaining the same geometric properties regardless of the coordinate system used.

Circle Center

The center of a circle in a coordinate plane is a fixed point from which every point on the circle is the same distance (the radius). In the rectangular form, (x-a)^2 + (y-b)^2 = r^2highlights that

• the center's coordinates are \((a, b)\)

For the specific exercise, once we convert the equation to the form (x-h)^2 + (y-k)^2 = r^2we recognize that the center is

• \((h, k)\).

This tells us that

• "h" and "k" are the coordinates of the circle's center in the rectangular plane.

Circle Radius

The radius of a circle is a critical measurement that indicates the distance from the center to any point on the circle. In the standard form of a circle's equation, (x-a)^2 + (y-b)^2 = r^2, "r" represents the radius. The exercise clearly shows that after converting from polar to rectangular, our equation takes this form, where

• r is resolved as 2.
This constant value confirms that every point on our circle is 2 units away from the center, allowing us to easily plot the circle in a rectangular coordinate system.