91Ó°ÊÓ

Let's start with completing the square, a crucial step in rewriting the circle equation correctly.
Initially, we have the equation: \[ x^{2} + y^{2} + 6x - 2y = 6 \]
To complete the square for both the x and y terms, we group them:
\[ x^{2} + 6x \] \[ y^{2} - 2y \]
For the x terms, take half of 6, which is 3, and then square it to get 9.
Add and subtract this 9 in the equation.
For the y terms, take half of -2, which is -1, and then square it to get 1.
Add and subtract this 1 as well.
Now add these constants (9 and 1) to the other side of the equation to keep it balanced:
\[ x^{2} + 6x + 9 + y^{2} - 2y + 1 = 6 + 9 + 1 \]
This simplifies to: \[ (x+3)^{2} + (y-1)^{2} = 16 \]
Completing the square transforms the given equation into a standard form that is much easier to interpret.

Determining the center and radius of a circle from its equation is fundamental.
From the equation we have: \[ (x+3)^{2} + (y-1)^{2} = 16 \]
We can identify the center \( (h, k) \) and the radius \( r \).
Notice the patterns:
\[ (x-h)^{2} + (y-k)^{2} = r^{2} \]
Here, \( h = -3 \) and \( k = 1 \). So, the center of the circle is at point \( (-3,1) \).
The right side of our equation gives us \( r^{2} = 16 \).
Taking the square root of 16, we find \( r = 4 \). So, the radius of the circle is 4 units.
Understanding how to extract the center and radius from the equation will help in both graph interpretation and geometric calculations.

Graphing involves transforming mathematical expressions onto a visual plane.
First, we identify the center: \[ (-3, 1) \]
Start by plotting this point on a coordinate graph.
From the center, use the radius of 4 units to mark a few key points.
These points should be 4 units away from the center both horizontally and vertically:

• To the left: \((-7,1)\)
• To the right: \((1,1)\)
• Upwards: \((-3,5)\)
• Downwards: \((-3,-3)\)

Draw the circle by connecting these points smoothly.
This graph visually represents the equation of the circle, making it easier to comprehend its spatial features.

The standard form of a circle is concise and informative.
Its general structure is: \[ (x-h)^{2} + (y-k)^{2} = r^{2} \]
\( (h, k) \) represents the circle's center coordinates.
\( r \) is the radius.
For our specific equation, \[ (x+3)^{2} + (y-1)^{2} = 16 \]
By comparing, we can see:

• \( h = -3 \)
• \( k = 1 \)
• \( r = \sqrt{16} = 4 \)

Using the standard form, we can effortlessly identify the circle's center and radius.
This clarity assists in graphing and solving related geometrical problems.
Mastering this form provides a solid foundation for working with circles in algebra and geometry.