Problem 62 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ (x+2)^{2}+(y-5)^{2}=16 $$

Step-by-Step Solution

Answer

The equivalent polar equation is \(r^2 + 4r \cos \theta - 10r \sin \theta = -13\).

Step 1: Recall Cartesian to Polar Conversion Formulas

We need to convert the given equation from Cartesian coordinates to polar coordinates. Recall that the conversion formulas are: - For the x-coordinate: \( x = r \cos \theta \)- For the y-coordinate: \( y = r \sin \theta \)- Also, \( r^2 = x^2 + y^2 \) can be useful to find the overall distance from the origin to a point (x, y).

Step 2: Substitute Cartesian Coordinatess

Substitute the expressions for \(x\) and \(y\) into the given equation:- \(x = r\cos\theta\)- \(y = r\sin\theta\)The given equation is \((x+2)^2+(y-5)^2 = 16\). Substitute these into the equation: \[(r \cos \theta + 2)^2 + (r \sin \theta - 5)^2 = 16\].

Step 3: Expand the Equation

Expand the equation to simplify it:\[(r \cos \theta + 2)^2 = r^2 \cos^2 \theta + 4r \cos \theta + 4\]\[(r \sin \theta - 5)^2 = r^2 \sin^2 \theta - 10r \sin \theta + 25\]Adding these expanded terms:\[r^2 \cos^2 \theta + 4r \cos \theta + 4 + r^2 \sin^2 \theta - 10r \sin \theta + 25 = 16\].

Step 4: Use Polar Identity

Apply the identity \( r^2 (\cos^2 \theta + \sin^2 \theta) = r^2 \). Hence, the equation becomes:\[r^2 + 4r \cos \theta - 10r \sin \theta + 29 = 16\].

Step 5: Simplify the Equation

Simplify this equation and solve for \(r\):\[r^2 + 4r \cos \theta - 10r \sin \theta + 29 = 16\]\[r^2 + 4r \cos \theta - 10r \sin \theta = -13\].This is the desired polar form of the equation.

Key Concepts

Cartesian to Polar Conversion

Understanding the shift from Cartesian to polar coordinates is crucial when working with equations describing geometric figures, like circles. Cartesian coordinates are based on the premise of defining positions on a plane using two perpendicular axes, typically labeled as "x" and "y." In contrast, polar coordinates represent points based on a distance from a reference point and an angle from a reference direction.

To undertake a Cartesian to Polar Conversion, you use a set of well-known mathematical relationships:

• The x-coordinate in a polar system is represented as \( x = r \cos \theta \), where \( r \) is the distance from the origin, and \( \theta \) is the angle.
• The y-coordinate turns into \( y = r \sin \theta \).
• An essential identity in these conversions is \( r^2 = x^2 + y^2 \), which represents the squared distance from the origin to any point \((x, y)\).

By applying these conversions to an equation like \((x+2)^2 + (y-5)^2 = 16\), we begin the shift from x and y descriptions to terms of \( r \) and \( \theta \), making it possible to redefine the nature of the equation in the polar system.

Polar Equations

After converting a Cartesian equation into polar format, you will work with polar equations that describe the same geometric feature but in terms of polar coordinates. The polar form of equations can be more intuitive for problems involving distance and angles from a central point.

Polar equations often come out looking different from their Cartesian counterparts due to their inherent reliance on trigonometric functions and identities. To illustrate, by replacing Cartesian coordinates in a circular equation like \((x+2)^2 + (y-5)^2 = 16\) with polar substitutions results in:
\[ (r \cos \theta + 2)^2 + (r \sin \theta - 5)^2 = 16 \]
Once expanded and simplified using identities like \( \cos^2 \theta + \sin^2 \theta = 1 \), the polar equation can highlight different properties or symmetries that are less obvious in Cartesian form. The goal often is to have a clean, concise representation, like resolving to an equation in the form \[ r^2 + 4r \cos \theta - 10r \sin \theta = -13 \], which speaks more directly in terms of r and \( \theta \).

Coordinate Transformation

Coordinate transformation refers to the process of changing the reference framework through which we interpret or solve geometrical problems. Transformations such as switching from Cartesian to polar coordinates allow for various insights and simplifications when dealing with certain symmetric shapes, like circles and spirals.

When performing a coordinate transformation, it's crucial to understand both the mechanical and conceptual shifts that occur:

• Mechanically, you are modifying the mathematical representation of points, such as transforming \( (x, y) \) coordinates into \( (r, \theta) \) using equations like \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Conceptually, you shift from a grid-based view of the plane to one that's angle and distance-based. This can simplify certain calculations, like those involving circles or periodic phenomena, as polar coordinates naturally fit the curvature and orientation.

In this transformation process, sometimes complex Cartesian equations simplify into more manageable forms, making it easier to read off the relevant geometrical information or apply specific analytical methods suited to polar-based systems.