To determine the equation of the new circle, let's follow a systematic approach to solve the problem:
1. Identify the center and radius of the given circle:
- The original equation of the circle is [tex]\(x^2 + y^2 - 8x - 6y + 24 = 0\)[/tex].
2. Convert the equation to the standard form:
- To convert the equation to the standard form, we complete the square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
- Start with the [tex]\(x\)[/tex] terms: [tex]\(x^2 - 8x\)[/tex]. To complete the square:
[tex]\[
x^2 - 8x = (x - 4)^2 - 16
\][/tex]
- Next, the [tex]\(y\)[/tex] terms: [tex]\(y^2 - 6y\)[/tex]. To complete the square:
[tex]\[
y^2 - 6y = (y - 3)^2 - 9
\][/tex]
- Combine these with the constant term from the original equation:
[tex]\[
x^2 + y^2 - 8x - 6y + 24 = 0
\][/tex]
becomes
[tex]\[
(x - 4)^2 - 16 + (y - 3)^2 - 9 + 24 = 0
\][/tex]
Simplifying this, we get:
[tex]\[
(x - 4)^2 + (y - 3)^2 - 1 = 0
\][/tex]
[tex]\[
(x - 4)^2 + (y - 3)^2 = 1
\][/tex]
- This represents a circle with center [tex]\((4, 3)\)[/tex] and radius [tex]\(1\)[/tex].
3. Formulate the equation of the new circle:
- We are given that the new circle has the same center [tex]\((4, 3)\)[/tex] but a different radius [tex]\(2\)[/tex].
- The standard form of the equation of a circle is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
- Substituting [tex]\(h = 4\)[/tex], [tex]\(k = 3\)[/tex], and [tex]\(r = 2\)[/tex], we get:
[tex]\[
(x - 4)^2 + (y - 3)^2 = 2^2
\][/tex]
4. Solution:
The equation that represents the described circle is:
[tex]\[
(x - 4)^2 + (y - 3)^2 = 2^2
\][/tex]
Thus, the correct equation from the choices provided is:
[tex]\[
(x - 4)^2 + (y - 3)^2 = 2^2
\][/tex]
