Problem 61 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ (x-3)^{2}+(y+1)^{2}=4 $$

Step-by-Step Solution

Answer

The polar equation is \(r^2 - 6r\cos\theta + 2r\sin\theta = -6\).

Step 1: Convert Cartesian to Polar Coordinates

The given Cartesian equation is \((x-3)^2 + (y+1)^2 = 4\). In polar coordinates, we have \(x = r\cos\theta\) and \(y = r\sin\theta\). Substituting these into the equation results in:\[(r\cos\theta - 3)^2 + (r\sin\theta + 1)^2 = 4\]

Step 2: Expand the Expression

Expand the equation from the previous step:\[(r\cos\theta - 3)^2 = r^2\cos^2\theta - 6r\cos\theta + 9\]\[(r\sin\theta + 1)^2 = r^2\sin^2\theta + 2r\sin\theta + 1\]

Step 3: Combine and Simplify

Combine the expanded expressions:\[r^2\cos^2\theta - 6r\cos\theta + 9 + r^2\sin^2\theta + 2r\sin\theta + 1 = 4\]Using the identity \(r^2(\cos^2\theta + \sin^2\theta) = r^2\), we simplify the equation:\[r^2 - 6r\cos\theta + 2r\sin\theta + 10 = 4\]

Step 4: Isolate Terms

Subtract 10 from both sides to isolate terms involving \(r\):\[r^2 - 6r\cos\theta + 2r\sin\theta = -6\]

Step 5: Final Polar Equation

The resulting polar equation is:\[r^2 - 6r\cos\theta + 2r\sin\theta = -6\] This is an equivalent polar representation of the original Cartesian equation.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a method of representing points in a plane using two numbers, essentially mapping points on a grid. This system uses two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical).

• A point in the Cartesian system is identified by a pair \( (x, y) \), where \( x \) is the distance from the y-axis and \( y \) is the distance from the x-axis.
• This coordinate system is very useful for plotting equations and understanding spatial relationships in a straightforward, intuitive way.
• Geometric shapes such as lines, circles, and parabolas can be easily described using Cartesian coordinates and equations.

In the provided exercise, we start with a Cartesian equation of a circle centered at the point \( (3, -1) \) with a radius of \( 2 \) since \( 4 = 2^2 \). This will be converted into polar coordinates, offering an alternative representation.

Polar Coordinates

Polar coordinates offer another way to locate a point on a plane, emphasizing the direction and distance from a fixed point, the origin. Unlike the Cartesian coordinates, which depend on perpendicular axes, polar coordinates use a radial system.

• A point in polar coordinates is represented by \( (r, \theta) \), where \( r \) is the radial distance from the origin and \( \theta \) is the angle measured from the positive x-axis.
• This system is particularly useful for dealing with problems involving circular and rotational symmetry.
• The relationship between Cartesian and polar coordinates is given by \( x = r\cos\theta \) and \( y = r\sin\theta \).

In the exercise solution, polar coordinates are employed to transform the given circle's Cartesian equation into a form that uses these radial and angular measurements, facilitating certain types of analysis and problem-solving in mathematics, especially calculus and trigonometry.

Equation Conversion

Equation conversion between coordinate systems can sometimes simplify the problem or provide a more insightful form of the equation. Converting from Cartesian to Polar coordinates involves substituting the expressions for \( x \) and \( y \) in terms of \( r \) and \( \theta \).

• Begin by substituting \( x = r\cos\theta \) and \( y = r\sin\theta \) into the Cartesian equation.
• Expand and simplify the resulting expression by applying trigonometric identities where possible.
• The aim is to express the equation solely in terms of \( r \) and \( \theta \), which can then depict geometric properties differently than their Cartesian counterparts.

The solution provided demonstrates the conversion process, resulting in a polar equation: \( r^2 - 6r\cos\theta + 2r\sin\theta = -6 \). This offers a deeper understanding of the circle's nature in terms of its radius and angle from the origin, a perspective that is often simpler for studying periodic functions or phenomena exhibiting rotational symmetry.