Problem 29 - The Calculus with Analytic Geometry

Question

Find a cartesian equation of the graph having the given polar equation.\(r=\frac{6}{2-3 \sin \theta}\)

Step-by-Step Solution

Answer

The Cartesian equation is: \[ 4x^2 - 5y^2 - 36y + 36 = 0 \]

Step 1: Convert the Polar Equation to Rectangular Coordinates

Start by noting that the given polar equation is in the form: \[ r = \frac{6}{2 - 3 \sin \theta} \] Next, remember the polar-to-rectangular coordinate transformations: \[ r^2 = x^2 + y^2 \] \[ r \sin \theta = y \] Plug in the values: \[ r = \frac{6}{2 - 3 \sin \theta} \] and \[ r \sin \theta = y \] so, \[ r = \frac{6}{2 - 3 \frac{y}{r}} \] Multiply through by the denominator: \[ r(2 - 3 \frac{y}{r}) = 6 \]

Step 2: Simplify the Equation

First, eliminate the fraction by multiplying both sides of the equation by the denominator: \[ 2r - 3y = 6 \] Next, substitute \( r = \sqrt{x^2 + y^2} \): \[ 2\sqrt{x^2 + y^2} - 3y = 6 \]

Step 3: Isolate the Square Root Term

Rearrange the equation to isolate the square root term: \[ 2\sqrt{x^2 + y^2} = 6 + 3y \]

Step 4: Square Both Sides of the Equation

To eliminate the square root, square both sides of the equation: \[ (2\sqrt{x^2 + y^2})^2 = (6 + 3y)^2 \] This gives: \[ 4(x^2 + y^2) = 36 + 36y + 9y^2 \]

Step 5: Simplify the Equation

Expand and simplify the result: \[ 4x^2 + 4y^2 = 36 + 36y + 9y^2 \] Combine like terms to obtain the final Cartesian equation: \[ 4x^2 - 5y^2 - 36y + 36 = 0 \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to determine a point's location on a plane. They are expressed as an ordered pair \((x, y)\). The x-coordinate reflects the horizontal distance from the origin, while the y-coordinate reflects the vertical distance. These coordinates are essential for graphing equations and performing various calculations in geometry and algebra.

The key advantages of using rectangular coordinates are:

• They provide a straightforward way to represent geometric figures and their positions.
• They simplify the process of identifying points and plotting them on a grid.

Understanding rectangular coordinates is crucial when converting from other coordinate systems, such as polar coordinates.

Polar Coordinates

Polar coordinates describe a point's location using the distance from a reference point (usually the origin) and an angle from a reference direction (usually the positive x-axis). They are expressed as \((r, \theta)\), where \r\ is the radial distance, and \theta\ is the angle.

Polar coordinates are particularly useful for:

• Describing points in circular or rotational symmetric scenarios.
• Simplifying certain types of integrals and solving differential equations.

This coordinate system allows for an alternative representation of points, which can be more intuitive for problems involving rotation or radial symmetry.

Coordinate Transformations

Coordinate transformations are essential for converting equations and points from one coordinate system to another. For instance, converting between polar coordinates and rectangular coordinates ensures we can analyze and interpret the information in a familiar Cartesian plane.

The conversion formulas are:

• \(r^2 = x^2 + y^2\) - converting from rectangular to polar coordinates.
• \(r \sin \theta = y\) - represents the y-coordinate in terms of polar coordinates.

They help perform operations such as integration and differentiation more comfortably, depending on the problem's nature.
For our case, using these transformations simplifies the process of finding a Cartesian equation for the given polar equation.

Equation Simplification

Equation simplification involves manipulating an equation into a more usable or interpretable form. In the given problem, we start with a polar equation and transform it into a Cartesian equation.
The simplification process typically includes steps such as:

• Substituting appropriate formulas to convert coordinates.
• Eliminating fractions or radicals by multiplying through or squaring.
• Combining like terms to consolidate the equation.

For example, starting with \( r = \frac{6}{2-3 \sin \theta}\), we used the transformations \( r \sin \theta = y \) and \( r = \sqrt{x^2 + y^2} \) to reframe the equation in terms of \x\ and \y\. This process highlights the importance of mathematical operations to simplify and bring equations into a more familiar format for further analysis.