To find the equation of a circle with a radius \( r \) and center at \( (h, v) \), we use the standard form of the equation of a circle. The standard form for the equation of a circle is:
[tex]\[ (x - h)^2 + (y - v)^2 = r^2 \][/tex]
Here’s a step-by-step explanation:
1. Identify the Components:
- Radius (r): The distance from the center of the circle to any point on the circle.
- Center (h, v): The coordinates of the center of the circle in the plane.
2. Substitute the Components into the Standard Form:
- The standard equation to express a circle in a Cartesian plane when the center is at \((h, v)\) and the radius is \( r \) is derived from the distance formula. This formula ensures that any point \((x, y)\) on the circle is exactly a distance \( r \) from the center \((h, v)\):
[tex]\[
(x - h)^2 + (y - v)^2 = r^2
\][/tex]
3. Check the Given Options:
- Option A: \( h^2 + v^2 = r^2 \): This is incorrect because this format doesn't contain variables for any arbitrary point \((x, y)\) on the circle.
- Option B: \( (x + h)^2 + (y + v)^2 = r^2 \): This option is incorrect because the signs inside the parentheses should be subtractive, not additive. Adding \( h \) and \( v \) would incorrectly shift the circle in the plane.
- Option C: \( (x - h)^2 + (y - v)^2 = r^2 \): This matches the correct format of the circle's equation, where \((x, y)\) is any point on the circle, and \((h, v)\) is the center.
- Option D: \( (x - v)^2 + (y - h)^2 = r^2 \): This option incorrectly swaps the coordinates of the center and the positional variables, which doesn't follow the standard equation form.
4. Conclusion:
- Comparing all the options, only Option C matches the standard form of the equation of a circle:
[tex]\[
(x - h)^2 + (y - v)^2 = r^2
\][/tex]
Therefore, the correct equation for a circle with a radius \( r \) and center at \( (h, v) \) is:
Option C: [tex]\( (x - h)^2 + (y - v)^2 = r^2 \)[/tex].
