Problem 53 - Precalculus

Question

Convert the equation from polar to rectangular form. Identify the resulting equation as a line, parabola, or circle. $$r^{2} \cos ^{2} \theta-2 r \cos \theta+r^{2} \sin ^{2} \theta=8$$

Step-by-Step Solution

Answer

The resulting equation is a circle centered at (1, 0) with radius 3.

Step 1: 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.

Step 2: 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 \[ x^2 - 2x + y^2 = 8 \].

Step 3: Rearrange the Equation

Notice that \( x^2 + y^2 \) is a circle equation format. Simplify the equation by keeping terms together: \( x^2 + y^2 - 2x = 8 \).

Step 4: Complete the Square

To complete the square for the \( x \) terms: \( x^2 - 2x \) can be rewritten as \( (x-1)^2 - 1 \). Substitute back: \[(x-1)^2 - 1 + y^2 = 8\].

Step 5: 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 \(r = 3\).

Step 6: Identify the Shape

The equation \((x-1)^2 + y^2 = 9 \) represents a circle centered at \((1, 0)\) with a radius of 3.

Key Concepts

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’s 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 θ.
• Coordinates are expressed in the form \( (r, \theta) \).

Polar coordinates are quite helpful for problems involving symmetry, such as those in circles and spirals. They provide an efficient way to express situations where relationships with angles and distances are more natural than traditional \( (x, y) \) coordinates.

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.

Rectangular coordinates are best for plotting straightforward graphs and performing tasks that involve basic arithmetic manipulations, like finding slopes or distances between points.

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’s 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.

This method is particularly useful in algebra and calculus for simplifying equations and identifying their geometric interpretations on a graph.

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’s 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.

Understanding circle equations is vital to analyzing curves and distances in space, making it a fundamental concept in geometry and algebra. This specific arrangement of the equation allows one to quickly identify the size and position of the circle based on its standard form.