To determine which equation represents the circle described in the question, let's follow these detailed steps:
### Step 1: Find the Center of the Given Circle
We are given the equation of a circle:
[tex]\[ x^2 + y^2 - 8x - 6y + 24 = 0 \][/tex]
We want to rewrite this equation in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex] by completing the square.
#### 1.1 - Group the variables:
[tex]\[ x^2 - 8x + y^2 - 6y = -24 \][/tex]
#### 1.2 - Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- For [tex]\(x\)[/tex]:
[tex]\[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[ y^2 - 6y \rightarrow (y - 3)^2 - 9 \][/tex]
Now substitute these into the equation:
[tex]\[ (x - 4)^2 - 16 + (y - 3)^2 - 9 = -24 \][/tex]
Simplify it:
[tex]\[ (x - 4)^2 + (y - 3)^2 - 25 = -24 \][/tex]
[tex]\[ (x - 4)^2 + (y - 3)^2 = 1 \][/tex]
From this, we can see that the center of the circle is:
[tex]\[ (h, k) = (4, 3) \][/tex]
### Step 2: Given Information
The radius of the new circle is [tex]\(2\)[/tex] units, and the center is the same as that of the original circle we calculated above.
### Step 3: Write the Equation of the New Circle
Using the center [tex]\((4, 3)\)[/tex] and the radius [tex]\(2\)[/tex], the equation of the new circle is:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 2^2 \][/tex]
This corresponds to:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 4 \][/tex]
### Step 4: Match with Given Choices
The options provided were:
1. [tex]\((x + 4)^2 + (y + 3)^2 = 2\)[/tex]
2. [tex]\((x - 4)^2 + (y - 3)^2 = 2\)[/tex]
3. [tex]\((x - 4)^2 + (y - 3)^2 = 2^2\)[/tex]
4. [tex]\((x + 4)^2 + (y + 3)^2 = 2^2\)[/tex]
The correct form of the equation for our new circle with center [tex]\((4, 3)\)[/tex] and radius [tex]\(2\)[/tex] is:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 2^2 \][/tex]
So, the correct choice is:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 2^2 \][/tex]
which simplifies to:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 4 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{(x - 4)^2 + (y - 3)^2 = 2^2} \][/tex]
