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The center-radius form of a circle is a very useful way to represent the equation of a circle. This form is given by\((x - h)^2 + (y - k)^2 = r^2\), where \( (h, k) \) represents the coordinates of the center of the circle and \( r \) represents the radius.
This format makes it easy to identify the key attributes of a circle.
When a circle's equation is in this form, you immediately know:

• The center, by looking at \(h\) and \(k\)
• The radius, by taking the square root of \(r^2\)

In our example, after completing the steps, the circle's equation is \((x + 5)^2 + (y - 7)^2 = 81\). Here, the center is \((-5, 7)\) and the radius is \(\sqrt{81} = 9\).

Completing the square is a method used to convert a quadratic equation into a perfect square trinomial. This technique is very useful in circle equations as it helps to bring the equation into the center-radius form.
To complete the square, follow these steps:

• Take the quadratic terms (e.g., \(x^2\) and \(y^2\))
• Group them with their respective linear terms (e.g., \(10x\) and \(-14y\))
• Find half of the linear coefficient, square it, and add it to both sides of the equation

In our exercise, we add 25 to both sides for the x terms and 49 for the y terms:
\((x^2 + 10x + 25) + (y^2 - 14y + 49) = 7 + 25 + 49\)
which simplifies to \((x + 5)^2 + (y - 7)^2 = 81\). This process transforms the equation into a form that easily reveals the circle's center and radius.

Understanding circle geometry is essential when dealing with circle equations. A circle is defined as the set of all points that are equidistant from a central point. This central point is called the center, and the distance from the center to any point on the circle is the radius.
There are a few important properties:

• All radii in a circle are equal
• The center of the circle is a fixed point
• The radius is the distance from this center to any point on the circle

In our example, the center is at \((-5, 7)\), meaning all points on the circle are 9 units away from this center point. The circle itself can be plotted easily once you know these two core characteristics.

The standard form of a circle is another way to write a circle's equation, and it leads into the center-radius form. The standard form is often written as \(x^2 + y^2 + Dx + Ey + F = 0\). To convert this into the center-radius form, you use the method of completing the square.
Starting with our standard form equation:
\( x^2 + y^2 + 10x - 14y - 7 = 0\), we complete the square for both x and y terms as shown in the provided solution.
After completing the square, we convert it to the center-radius form of \((x + 5)^2 + (y - 7)^2 = 81\). This conversion is vital because it allows us to easily extract the circle's center and radius, making both geometric and algebraic interpretations straightforward.