%5E2%2B(y-1)%5E2%3D9.jpeg "(x-6)^2+(y-1)^2=9")
Exploring the Circle: (x-6)^2 + (y-1)^2 = 9
This equation represents a circle in the standard form. Let's break down its components and understand what it tells us about the circle.
The Standard Form Equation
The standard form equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
Analyzing the Equation: (x - 6)^2 + (y - 1)^2 = 9
Comparing this equation to the standard form, we can identify the following:
- Center: (h, k) = (6, 1)
- Radius: r^2 = 9, therefore r = 3
Understanding the Circle
This equation describes a circle with:
- Center: Located at the point (6, 1) on the coordinate plane.
- Radius: 3 units.
This means that every point on the circle is exactly 3 units away from the point (6, 1).
Visual Representation
Imagine drawing a circle on a graph. You would place the center at the point (6, 1) and then draw a circle with a radius of 3 units. This circle would encompass all points that are 3 units away from the center.
Key Points to Remember
- The standard form equation provides a direct way to find the center and radius of a circle.
- Understanding the relationship between the equation and the circle's properties is essential for analyzing and manipulating circles in various contexts.