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Magazine Chapter 7: Problem 53
Convert the equation from polar to rectangular form. Identify the resulting equation as a line, parabola, or circle. $$r^{2} \cos ^{2} \theta-2 r \cos \theta+r^{2} \sin ^{2} \theta=8$$
Short Answer
Expert verified
The resulting equation is a circle centered at (1, 0) with radius 3.
Step by step solution
01
Recall Polar and Rectangular Relationships
In polar coordinates, the relationships with rectangular coordinates are: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). We'll use these to convert the equation.
02
Substitute Polar Relationships
Replace \( r \cos \theta \) with \( x \) and \( r^2 \) with \( x^2 + y^2 \) in the given equation: \[ r^2 \cos^2 \theta - 2r \cos \theta + r^2 \sin^2 \theta = 8 \] becomes \[ x^2 - 2x + y^2 = 8 \].
03
Rearrange the Equation
Notice that \( x^2 + y^2 \) is a circle equation format. Simplify the equation by keeping terms together: \( x^2 + y^2 - 2x = 8 \).
04
Complete the Square
To complete the square for the \( x \) terms: \( x^2 - 2x \) can be rewritten as \( (x-1)^2 - 1 \). Substitute back: \[(x-1)^2 - 1 + y^2 = 8\].
05
Final Formulation
Add 1 to both sides to simplify: \[(x-1)^2 + y^2 = 9 \]. This is now in the standard form of a circle \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k) = (1, 0)\) and \(r = 3\).
06
Identify the Shape
The equation \((x-1)^2 + y^2 = 9 \) represents a circle centered at \((1, 0)\) with a radius of 3.
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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 offer a unique way of representing points on a plane, typically using a pair of measurements: a distance from a reference point and an angle from a reference direction. Here鈥檚 a bit more on how they work:
- The reference point is known as the origin, similar in concept to the origin in Cartesian coordinates.
- Angles are measured from a reference direction, usually the positive x-axis.
- The distance from the origin to a point in the plane is often denoted by r and the angle, typically in radians, is represented as 胃.
- Coordinates are expressed in the form \( (r, \theta) \).
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, use two numbers to pinpoint a location on a two-dimensional plane. The basic setup looks like this:
- The horizontal axis, or x-axis, and the vertical axis, or y-axis, intersect at the origin, represented by the coordinates \( (0, 0) \).
- Any point in the space is represented by a pair \( (x, y) \), where \( x \) denotes the horizontal distance from the origin, and \( y \) denotes the vertical distance from the origin.
- These coordinates are extremely useful for describing geometric shapes and equations precisely.
Completing the Square
Completing the square is a technique used to transform quadratic expressions into a perfect square trinomial, aiding in solving equations and graphing conics such as circles. Here鈥檚 a brief guide on how this method works:
- Given a quadratic expression \( ax^2 + bx \), add and subtract the square of half of the coefficient of \( x \).
- For example, in \( x^2 - 2x \), halve \( -2 \) to get \( -1 \) and square it to get \( 1 \). Add and subtract \( 1 \) to form \( (x - 1)^2 \).
- Rewriting the expression as a square makes it easier to analyze graphs or integrate into larger equations, like those of circle equations.
Circle Equation
The equation of a circle in rectangular coordinates has the form \( (x - h)^2 + (y - k)^2 = r^2 \), which effortlessly captures its geometric characteristics:
- Here, \( (h, k) \) denote the coordinates of the circle鈥檚 center.
- The radius of the circle is represented by \( r \), describing the distance from the center to any point on the circle itself.
- A circle is symmetric around its center, making it quite distinct among conics in its uniformity.
