Let's start with the given equation of the circle:
[tex]\[
(x - 2)^2 + (y - 3)^2 = 25
\][/tex]
From this equation, we can identify the center of the circle, [tex]\((h, k)\)[/tex], and the radius [tex]\(r\)[/tex]. The standard form of a circle's equation is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
By comparing, we can see that the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((2, 3)\)[/tex], and the radius [tex]\(r\)[/tex] is [tex]\(\sqrt{25} = 5\)[/tex].
Next, we need to find the result of shifting the circle to the left by 2 units. When we shift a circle horizontally, we only change the [tex]\(x\)[/tex]-coordinate of the center. Specifically:
1. To shift left 2 units: we subtract 2 from the [tex]\(x\)[/tex]-coordinate.
Therefore, the new center of the circle will be:
- New [tex]\(x\)[/tex]-coordinate: [tex]\(2 - 2 = 0\)[/tex]
- [tex]\(y\)[/tex]-coordinate remains unchanged at [tex]\(3\)[/tex].
So, the new center of the circle is [tex]\((0, 3)\)[/tex].
Now, let's determine which of the given options correctly describes this translation:
- A. Both the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates of the center of the circle increase by 2. (Incorrect, since only the [tex]\(x\)[/tex]-coordinate changed, and it decreased, not increased.)
- B. Both the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates of the center of the circle decrease by 2. (Incorrect, since the [tex]\(y\)[/tex]-coordinate did not change.)
- C. The [tex]\(x\)[/tex]-coordinate of the center of the circle decreases by 2. (Correct, the [tex]\(x\)[/tex]-coordinate changed from 2 to 0, a decrease by 2 units.)
- D. The [tex]\(y\)[/tex]-coordinate of the center of the circle decreases by 2. (Incorrect, since the [tex]\(y\)[/tex]-coordinate remained at 3.)
Thus, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
