To find the equation of the new circle with a radius of 2 units and the same center as the circle with the equation [tex]\( x^2 + y^2 - 8x - 6y + 24 = 0 \)[/tex], we need to follow these steps:
### Step 1: Find the Center of the Given Circle
To determine the center of the circle described by the equation [tex]\( x^2 + y^2 - 8x - 6y + 24 = 0 \)[/tex], we can complete the square for the terms involving [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Rearrange the equation to group the [tex]\( x \)[/tex]-terms and [tex]\( y \)[/tex]-terms:
[tex]\[ x^2 - 8x + y^2 - 6y = -24. \][/tex]
2. Complete the square for [tex]\( x \)[/tex]:
- Take the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-8\)[/tex]), divide by 2 to get [tex]\(-4\)[/tex], and square it to get [tex]\( 16 \)[/tex].
- Add and subtract [tex]\( 16 \)[/tex] inside the equation.
[tex]\[ x^2 - 8x + 16 + y^2 - 6y = -24 + 16. \][/tex]
3. Complete the square for [tex]\( y \)[/tex]:
- Take the coefficient of [tex]\( y \)[/tex] (which is [tex]\(-6\)[/tex]), divide by 2 to get [tex]\(-3\)[/tex], and square it to get [tex]\( 9 \)[/tex].
- Add and subtract [tex]\( 9 \)[/tex] inside the equation.
[tex]\[ x^2 - 8x + 16 + y^2 - 6y + 9 = -24 + 16 + 9. \][/tex]
4. Simplify the equation:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 1. \][/tex]
Thus, the center of the circle is [tex]\( (4, 3) \)[/tex] and its radius is [tex]\(\sqrt{1} = 1\)[/tex].
### Step 2: Write the Equation of the New Circle
The new circle has the same center as the above circle, which is [tex]\( (4, 3) \)[/tex], and a radius of 2 units.
The general equation of a circle with center [tex]\( (h, k) \)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2. \][/tex]
Substitute [tex]\( h = 4 \)[/tex], [tex]\( k = 3 \)[/tex], and [tex]\( r = 2 \)[/tex]:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 2^2. \][/tex]
Simplify the radius squared:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 4. \][/tex]
Therefore, the equation representing the new circle is:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 4. \][/tex]
Among the given options, this corresponds to the equation:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 2^2. \][/tex]
