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Magazine Chapter 8: Problem 57
Convert to a polar equation. $$x^{2}-2 x+y^{2}=0$$
Short Answer
Expert verified
r = 2 \, cos \theta
Step by step solution
01
Identify Cartesian Equation
Start with the given Cartesian equation: \[ x^{2} - 2x + y^{2} = 0 \]
02
Complete the Square
Rewrite the equation by completing the square for the x terms.First, recall that completing the square involves the expression \( x^2 - 2x \). To complete the square, add and subtract 1: \[ x^2 - 2x + 1 - 1 + y^2 = 0 \]This simplifies to: \[ (x - 1)^2 - 1 + y^2 = 0 \]Then move the -1 to the right side: \[ (x - 1)^2 + y^2 = 1 \]
03
Convert to Polar Coordinates
Recall the conversion formulas from Cartesian to Polar coordinates: \[ x = r \, cos \theta \] \[ y = r \, sin \theta \] The given equation has been completed to: \[ (x - 1)^2 + y^2 = 1 \] Substitute the polar coordinates into this equation: \[ (r \, cos \theta - 1)^2 + (r \, sin \theta)^2 = 1 \]
04
Simplify Polar Equation
Expand and simplify the polar coordinate equation:\[ (r \, cos \theta - 1)^2 + (r \, sin \theta)^2 = 1 \]This becomes:\[ (r \, cos \theta)^2 - 2r \, cos \theta + 1 + (r \, sin \theta)^2 = 1 \]Combine like terms using the Pythagorean identity \( cos^2 \theta + sin^2 \theta = 1 \):\[ r^2 (cos^2 \theta + sin^2 \theta) - 2r \, cos \theta + 1 = 1 \]Simplify to:\[ r^2 - 2r \, cos \theta + 1 = 1 \]Subtract 1 from both sides: \[ r^2 - 2r \, cos \theta = 0 \]
05
Factorize
Factor out r:\[ r (r - 2 \, cos \theta) = 0 \]Therefore, the polar equation is:\[ r = 0 \] or \[ r = 2 \, cos \theta \]Since \(r = 0\) represents the origin, the polar equation simplifies to \( r = 2 \, cos \theta \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Coordinates
In mathematics, polar coordinates provide a different way of representing points in a plane. Instead of using the usual Cartesian coordinates \((x, y)\), we use a distance from the origin called \(r\) and an angle \(\theta\) from the positive x-axis.
**Key Points About Polar Coordinates:**
For example, in polar form, a point could be written as \((r, \theta)\). Converting between Cartesian and polar coordinates involves specific formulas.
**Key Points About Polar Coordinates:**
- **\(r\)**: This is the radius, the distance from the origin to the point.
- **\(\theta\)**: This is the angle measured from the positive x-axis to the line connecting the origin to the point.
For example, in polar form, a point could be written as \((r, \theta)\). Converting between Cartesian and polar coordinates involves specific formulas.
Completing the Square
Completing the square is a crucial method in algebra used for solving quadratic equations and transforming them into a perfect square form.
**Steps to Complete the Square:**
**Steps to Complete the Square:**
- Identify the quadratic and linear terms in the equation. For example, in \(x^2 - 2x + y^2 = 0\), focus on \(x^2 - 2x\).
- Take half of the linear term coefficient, square it, and add/subtract that square. For \(x^2 - 2x\), half of -2 is -1 and squaring -1 gives us 1.
- Add and subtract this square inside the equation to form a perfect square trinom in the equation: \(x^2 - 2x + 1 - 1 + y^2 = 0\).
- Rearrange the equation: \((x - 1)^2 + y^2 = 1\).
Conversion Formulas
Conversion between Cartesian and polar coordinates requires specific formulas.
**Formulas To Convert Cartesian Coordinates to Polar Coordinates:**
**Using These Formulas:**
Given a Cartesian equation, substitute \(x\) and \(y\) with their polar forms:
In our step-by-step solution, we started with \((x - 1)^2 + y^2 = 1\), then substituted \(x\) and \(y\) to get \((r \cos \theta - 1)^2 + (r \sin \theta)^2 = 1\). Simplifying this using trigonometric identities led to our final form: \(r = 2 \cos \theta\). This is the converted polar equation.
**Formulas To Convert Cartesian Coordinates to Polar Coordinates:**
- **For x:** \( x = r \cos \theta \)
- **For y:** \( y = r \sin \theta \)
**Using These Formulas:**
Given a Cartesian equation, substitute \(x\) and \(y\) with their polar forms:
In our step-by-step solution, we started with \((x - 1)^2 + y^2 = 1\), then substituted \(x\) and \(y\) to get \((r \cos \theta - 1)^2 + (r \sin \theta)^2 = 1\). Simplifying this using trigonometric identities led to our final form: \(r = 2 \cos \theta\). This is the converted polar equation.
