To determine the correct equation for a circle with radius [tex]\( r \)[/tex] and center at [tex]\( (h, k) \)[/tex], we need to recall the general form of the equation for a circle.
The standard equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is given by:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Let's analyze each of the provided options to find the one that matches this standard form.
A. [tex]\((x - k)^2 + (y - h)^2 = r^2\)[/tex]
This equation incorrectly switches the variables [tex]\( h \)[/tex] and [tex]\( k \)[/tex] as coordinates in the equation. The correct form doesn't swap these variables; it should have the form [tex]\((x - h)\)[/tex] and [tex]\((y - k)\)[/tex].
B. [tex]\(h^2 + k^2 = r^2\)[/tex]
This equation does not represent a circle’s equation in Cartesian coordinates. It looks like a Pythagorean theorem but does not fit the standard form of a circle.
C. [tex]\((x + h)^2 + (y + k)^2 = r^2\)[/tex]
This option incorrectly includes addition instead of subtraction. The standard form includes the subtraction of the center coordinates from the x and y coordinates, respectively.
D. [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]
This matches the standard form exactly, with the correct subtraction of [tex]\( h \)[/tex] from [tex]\( x \)[/tex] and [tex]\( k \)[/tex] from [tex]\( y \)[/tex].
Thus, the correct equation for a circle with radius [tex]\( r \)[/tex] and center at [tex]\( (h, k) \)[/tex] is:
[tex]\[
\boxed{(x - h)^2 + (y - k)^2 = r^2}
\][/tex]
So, the correct answer is option D.
