Problem 87 - Algebra and Trigonometry

Question

Show that the graph of the equation \(r=2 a \cos \theta, a>0,\) is a circle of radius \(a\) with center \((a, 0)\) in rectangular coordinates.

Step-by-Step Solution

Answer

The graph of the equation is a circle with center \((a, 0)\) and radius \(a\).

Step 1: Write the given polar equation

Start with the given polar equation: \[ r = 2a \, \cos \theta \].

Step 2: Use polar to rectangular coordinate conversion

Recall the conversions between polar and rectangular coordinates: \[ x = r \cos \theta \] and \[ y = r \sin \theta \]. We also have \[ r^2 = x^2 + y^2 \].

Step 3: Multiply both sides by \( r \)

Multiply both sides of the equation \( r = 2a \cos \theta \) by \( r \): \[ r^2 = 2a r \cos \theta \].

Step 4: Substitute rectangular coordinates expressions

Substitute \( r^2 = x^2 + y^2 \) and \( r \cos \theta = x \) into the equation: \[ x^2 + y^2 = 2ax \].

Step 5: Rearrange the equation

Rearrange the equation to represent the standard form of a circle's equation: \[ x^2 - 2ax + y^2 = 0 \].

Step 6: Complete the square

Complete the square for the \( x \)-terms: \[ (x - a)^2 - a^2 + y^2 = 0 \].

Step 7: Simplify to the standard form of a circle

Add \( a^2 \) to both sides to isolate the squared term: \[ (x - a)^2 + y^2 = a^2 \]. This is the standard form of a circle's equation with center \((a, 0)\) and radius \(a\).

Key Concepts

polar coordinates

Polar coordinates are a way to describe the position of a point in a plane using two values: the distance from a reference point (usually the origin) and the angle from a reference direction (usually the positive x-axis).
The coordinate pair \(r, \theta\) defines a point in the polar coordinate system.
Here, \(r\) is the radius or distance from the origin, and \(\theta\) is the angle measured in radians from the positive x-axis.
This system is particularly useful for problems featuring circular or spiral patterns and can make calculations simpler compared to Cartesian coordinates.
For example, the point with polar coordinates \(5, \pi/4\) is 5 units away from the origin, forming an angle of \(\pi/4\) radians with the x-axis.

rectangular coordinates

Rectangular coordinates (or Cartesian coordinates) describe a point's position using two perpendicular axes: the x-axis and the y-axis.
A point on the plane is represented as \(x, y\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.
To convert from polar to rectangular coordinates, use the following relationships:

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

Conversely, to convert from rectangular to polar coordinates, use:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan \left( \frac{y}{x} \right)\)

Understanding these conversions is crucial for moving between coordinate systems.

circle equation

In rectangular coordinates, the standard equation for a circle centered at point \(h, k\) with a radius \(r\) is: \((x - h)^2 + (y - k)^2 = r^2\)
From this general form, you can see that the circle's center is \(h, k\), and its radius is \(r\).
To show that an equation represents a circle, you often need to manipulate the equation into this standard form.
In our solution, the given polar equation \(r = 2a \cos \theta\) was transformed into its rectangular equivalent.
By converting the polar equation to rectangular coordinates, we eventually arrived at the equation \((x - a)^2 + y^2 = a^2\).
This equation matches the standard form of a circle with center \(a, 0\) and radius \(a\), confirming that the graph is indeed a circle.