The standard equation of a circle with its center precisely at the origin (0,0) of a coordinate plane is x² + y² = r².
Deconstructing the Equation
This elegant formula describes the relationship between any point (x, y) on the circle's circumference and its constant distance from the origin. Understanding each component is key to grasping the equation's meaning:
- x: Represents the x-coordinate of any given point that lies on the circle.
- y: Represents the y-coordinate of the same given point that lies on the circle.
- r: Denotes the radius of the circle, which is the fixed distance from the center (in this case, the origin) to any point on its circumference.
- r²: Is the square of the radius. This value is always positive and determines the size of the circle.
Why This Form?
This particular form of the circle equation is a direct application of the distance formula, which itself is derived from the Pythagorean theorem. For any point (x, y) on the circle, the horizontal distance from the origin to the point is x, and the vertical distance is y. The distance from the origin (0, 0) to that point (x, y) is precisely the radius, r. When you visualize this as a right-angled triangle with legs x and y and hypotenuse r, the Pythagorean theorem states: leg₁² + leg₂² = hypotenuse², which translates directly to x² + y² = r².
Practical Examples
Let's explore a few examples to illustrate how this equation works in real-world scenarios:
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Example 1: Circle with a Known Radius
- If a circle is centered at the origin and has a radius of 6 units, its standard equation would be:
x² + y² = 6²
x² + y² = 36
- If a circle is centered at the origin and has a radius of 6 units, its standard equation would be:
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Example 2: Finding the Equation from a Point
- Suppose a circle centered at the origin passes through the point (-3, 4). To find its equation, you first need to determine the radius squared (r²):
r² = (-3)² + 4²
r² = 9 + 16
r² = 25
Therefore, the standard equation of the circle is:
x² + y² = 25 (Here, the radius r is 5)
- Suppose a circle centered at the origin passes through the point (-3, 4). To find its equation, you first need to determine the radius squared (r²):
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Example 3: Verifying if a Point is on the Circle
- Consider the circle with the equation x² + y² = 100. Is the point (6, 8) on this circle?
Substitute the coordinates into the equation:
6² + 8² = 36 + 64 = 100
Since 100 = 100, the point (6, 8) is on the circle.
- Consider the circle with the equation x² + y² = 100. Is the point (6, 8) on this circle?
Key Characteristics
The following table summarizes the fundamental characteristics of a circle centered at the origin:
| Characteristic | Description |
|---|---|
| Center | (0, 0) – Always located at the origin of the coordinate system |
| Radius | r – The constant distance from the center to any point on the circle's edge |
| Equation | x² + y² = r² |
| Components | x-coordinate, y-coordinate, and the radius squared |
For a more comprehensive understanding of circle equations, including circles not centered at the origin, you can refer to additional resources such as Khan Academy's explanation of the equation of a circle.
[[Circle Equation]]
