Chapter 10: Problem 87
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.
Short Answer
Expert verified
The graph of the equation is a circle with center \((a, 0)\) and radius \(a\).
Step by step solution
01
Write the given polar equation
Start with the given polar equation: \[ r = 2a \, \cos \theta \].
02
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 \[ r^2 = x^2 + y^2 \].
03
Multiply both sides by \( r \)
Multiply both sides of the equation \( r = 2a \cos \theta \) by \( r \): \[ r^2 = 2a r \cos \theta \].
04
Substitute rectangular coordinates expressions
Substitute \( r^2 = x^2 + y^2 \) and \( r \cos \theta = x \) into the equation: \[ x^2 + y^2 = 2ax \].
05
Rearrange the equation
Rearrange the equation to represent the standard form of a circle's equation: \[ x^2 - 2ax + y^2 = 0 \].
06
Complete the square
Complete the square for the \( x \)-terms: \[ (x - a)^2 - a^2 + y^2 = 0 \].
07
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 \(a\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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:
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:
- \(x = r \cos \theta \)
- \(y = r \sin \theta \)
- \(r = \sqrt{x^2 + y^2}\)
- \(\theta = \arctan \left( \frac{y}{x} \right)\)
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.
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.