91Ó°ÊÓ

In coordinate geometry, every circle can be defined by its center and its radius. The center is a specific point on the coordinate plane, while the radius is the constant distance from the center to any point on the circle.
Let's break down our specific problem. The given center is \( (8, -11) \), and the radius is 5 units. This means every point on the circle is exactly 5 units away from \( (8, -11) \).
To find the equation of a circle, you need to identify:

• The center, noted as \( (h, k) \).
• The radius, noted as \( r \).

Once you have these two pieces of information, you can plug them into the standard circle equation form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
For this exercise, \( h = 8 \), \( k = -11 \,\text{ and }r = 5 \). This leads us to the specific equation form:
\[ (x - 8)^2 + (y + 11)^2 = 25 \]

The distance formula is essential in coordinate geometry for calculating the distance between two points on the Cartesian plane. It's derived from the Pythagorean theorem. The formula for the distance between two points, (x1, y1) and (x2, y2), is:
\[ \text{distance} = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
In the given problem, we used this concept to recognize that all points on the circle are 5 units away from the center (8, -11). This matches the definition given by the circle equation: \[ (x - 8)^2 + (y + 11)^2 = 25 \]
The term '25' here stems from squaring the radius (5), ensuring that every point on this circle maintains a fixed distance (radius) from the center. Understanding this distance ensures we get the correct equation for any given circle.

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to discuss geometric figures in the coordinate plane.
A circle in coordinate geometry isn't just a shape; it's represented as a set of points that meet a specific algebraic condition. This condition is determined by the circle's center and radius, fitting into the standard equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Where:

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

With the given problem, our use of coordinate geometry helps in translating the geometric concept of a circle into a tangible equation:
\[ (x - 8)^2 + (y + 11)^2 = 25 \]
This ability to use equations to represent and manipulate geometric figures is a powerful tool in mathematics, enhancing our understanding and providing solutions to complex problems efficiently.