To determine the equation for a circle with a radius [tex]\( r \)[/tex] and center at the point [tex]\((h, v)\)[/tex], we should recall the standard form of the equation of a circle.
The general equation for a circle in a two-dimensional coordinate system is:
[tex]\[
(x - h)^2 + (y - v)^2 = r^2
\][/tex]
Here:
- [tex]\( (h, v) \)[/tex] is the center of the circle.
- [tex]\( r \)[/tex] is the radius of the circle.
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are the coordinates of any point on the circle.
Given these parameters, let's compare this form with the provided options:
A. [tex]\((x-y)^2+(y-h)^2=r^2\)[/tex]
- This does not match the standard form. The transformations inside the parentheses do not correspond to the correct variables for the center of the circle.
B. [tex]\(\hbar^2+v^2=t^2\)[/tex]
- This is not even an equation related to a circle. It uses different variables that do not fit our circle's equation structure.
C. [tex]\((x+h)^2+(y+v)^2=r^2\)[/tex]
- This equation incorrectly places additions inside the parentheses. According to the standard form, the center coordinates should be subtracted, not added.
D. [tex]\((x-h)^2+(y-v)^2=r^2\)[/tex]
- This correctly follows the standard form. The center [tex]\((h, v)\)[/tex] and the radius [tex]\( r \)[/tex] are placed correctly.
Based on this analysis, the equation that accurately represents a circle with a radius of [tex]\( r \)[/tex] and center at [tex]\((h, v)\)[/tex] is:
[tex]\[
(x - h)^2 + (y - v)^2 = r^2 \
Which shows that the correct answer is:
D. \((x-h)^2+(y-v)^2=r^2\)
Therefore, the correct choice is:
\[
\boxed{4}
\][/tex]
