91Ó°ÊÓ

The circle equation provides a way to describe a circle within the Cartesian plane. The general form of a circle's equation is \[(x - h)^2 + (y - k)^2 = r^2\]where

• \( (h, k) \) represents the center of the circle
• \( r \) represents the radius

In this exercise, we transform the polar equation given as \( r = a \cos \theta + b \sin \theta \) into the Cartesian form by expressing it in terms of \( x \) and \( y \). This involves using the equations of transformation that link polar coordinates to Cartesian coordinates, such as:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

By converting the polar equation, we reach the Cartesian form \( r^2 = ax + by \). This then leads us to the equation \((x - \frac{a}{2})^2 + (y - \frac{b}{2})^2 = \frac{a^2}{4} + \frac{b^2}{4}\), identifying the circle's center and radius.

Cartesian coordinates provide a straightforward system to graphically display points within a plane. They consist of an \( x \) value and a \( y \) value, forming a unique ordered pair \((x, y)\) that represents a specific location. This coordinate system is built on perpendicular axes:

• The horizontal axis, denoted by \( x \), is commonly referred to as the abscissa.
• The vertical axis, denoted by \( y \), is termed the ordinate.

Using Cartesian coordinates simplifies translating polar graphs (such as those in polar form \(r = a \cos \theta + b \sin \theta\)) into recognizable shapes. Converting from polar coordinates can sometimes seem complex but is manageable by integrating the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\).
This conversion enables us to express the relationship \( r = a \cos \theta + b \sin \theta \) as \( r^2 = x^2 + y^2 = ax + by \) in rectangular terms. Simplifying this further helps visualize the circle, identifying its properties like center and radius more readily.

Completing the square is a useful algebraic method for transforming a quadratic equation into a form that reveals specific features of conics, such as ellipses and circles. For our exercise, it is applied to transform the terms \(x^2 - ax\) and \(y^2 - by\) in our equation process.To complete the square for a term like \(x^2 - ax\):

• Take half of the coefficient of \(x\), which is \(\frac{a}{2}\),
• Square it, yielding \((\frac{a}{2})^2\),
• Add and subtract this square inside the equation: \((x - \frac{a}{2})^2 - (\frac{a}{2})^2\).

By repeating this step for \(y^2 - by\), we get the equation in a neat completed square form as:\[(x - \frac{a}{2})^2 + (y - \frac{b}{2})^2 = (\frac{a}{2})^2 + (\frac{b}{2})^2\]This form clearly shows:

• The center of the circle at \((\frac{a}{2}, \frac{b}{2})\),
• The radius as \(\sqrt{(\frac{a}{2})^2 + (\frac{b}{2})^2}\).

Completing the square helps not only in rearranging equations but also in visualizing the geometrical figures represented by those equations.