The standard form of the equation of a circle is expressed as:
(x - h)² + (y - k)² = r²
In this equation:
- (h, k) represents the center of the circle.
- r denotes the radius of the circle.
To write the standard form, you need to determine the center and radius of the circle. Here’s a step-by-step process:
- Identify the Center: The coordinates of the center are given as (h, k). If you’re provided with the center, simply note these values.
- Determine the Radius: The radius is the distance from the center to any point on the circle. If you know a point on the circle, you can calculate the radius using the distance formula:
r = sqrt((x - h)² + (y - k)²)
- Substitute values: Once you have both the center (h, k) and the radius r, plug these values into the standard equation:
(x - h)² + (y - k)² = r²
Example:
Suppose you’re given the center (3, -2) and a point on the circle (6, -2). To find the radius:
- Using the distance formula:
r = sqrt((6 - 3)² + (-2 - (-2))²) = sqrt(3² + 0²) = 3
- Now, substitute the values into the standard form:
(x - 3)² + (y + 2)² = 3²
This simplifies to:
(x - 3)² + (y + 2)² = 9
And that’s how you write the equation of a circle in standard form! Remember, understanding the characteristics of the circle is key to expressing its equation correctly.