91Ó°ÊÓ

Understanding the standard form equation of a circle is crucial to solving circle-related problems.
The standard form equation for a circle with given center coordinates \((h, k)\) and radius \(r\) is:
\((x - h)^2 + (y - k)^2 = r^2\).
This equation tells us that any point \((x, y)\) on the circle is at a fixed distance, the radius, from the center of the circle.
The terms \( (x - h)\) and \( (y - k)\) represent the horizontal and vertical distances from the center to any point on the circle, respectively. Squaring these distances helps in converting the circle's equation in a format that we can easily manipulate.
When working with a circle's equation, always start by identifying the center \((h, k)\) and the radius \(r\) to easily plug them into this standard form equation.

The center of a circle is a vital piece of information needed to write the circle's equation.
In our exercise example, the center coordinates are given as \((-1, -6)\).
When substituting these coordinates into the standard form equation, replace \(h\) and \(k\) with -1 and -6, respectively.
This transforms the equation from \((x - h)^2 + (y - k)^2 = r^2\) to \((x - (-1))^2 + (y - (-6))^2 = r^2\).
Simplifying the double negatives, it becomes \((x + 1)^2 + (y + 6)^2 = r^2\).
Always remember to simplify the equation by handling negative signs correctly, especially when substituting values.

Once we've substituted the center coordinates into the standard form equation, the next step involves the radius.
For our problem, the radius is given as \(\frac{1}{3}\).
The standard form equation after substituting the center coordinates looks like this:
\((x + 1)^2 + (y + 6)^2 = r^2\).
Now, substitute \(r = \frac{1}{3}\) into the equation.
Since the radius term in the standard form equation is squared, we need to square the radius value:
\((\frac{1}{3})^2 = \frac{1}{9}\).
This changes our equation to: \((x + 1)^2 + (y + 6)^2 = \frac{1}{9}\).
This final equation \((x + 1)^2 + (y + 6)^2 = \frac{1}{9}\) represents the full equation of the circle with the given center and radius.