What is the standard form of a circle's equation?
Understand the Problem
The question is asking for the standard form of a circle's equation, which is a fundamental concept in geometry related to the relationship between a circle's center and radius.
Answer
The standard form of a circle's equation is $ (x - h)^2 + (y - k)^2 = r^2 $.
Answer for screen readers
The standard form of a circle's equation is given by:
$$(x - h)^2 + (y - k)^2 = r^2$$
Steps to Solve
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Identify the circle's center and radius
You need to determine the center of the circle and the radius. In the standard form of a circle's equation, which is given by:
$$ (x - h)^2 + (y - k)^2 = r^2 $$
Here, $(h, k)$ is the center and $r$ is the radius. -
Substitute the values into the formula
Replace $h$, $k$, and $r$ in the standard form equation with the respective values. For example, if the center is at $(3, -2)$ and the radius is $5$, the equation would become:
$$ (x - 3)^2 + (y + 2)^2 = 5^2 $$ -
Simplify the equation
Finally, calculate $r^2$ and simplify the equation. Continuing the example:
$$ (x - 3)^2 + (y + 2)^2 = 25 $$
The standard form of a circle's equation is given by:
$$(x - h)^2 + (y - k)^2 = r^2$$
More Information
The standard form of a circle's equation is very useful in geometry and analytic geometry. It helps you determine critical properties of the circle such as its center and radius quickly.
Tips
- Misidentifying the center: Make sure to correctly identify $(h, k)$ from the coordinates provided.
- Incorrectly calculating $r^2$: Always square the radius correctly. For example, if the radius is 4, $r^2$ must be 16, not 12 or any other number.
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