Which equation represents the circle described?

- The radius is 2 units.
- The center is the same as the center of a circle whose equation is [tex]x^2 + y^2 - 8x - 6y + 24 = 0[/tex].

A. [tex](x + 4)^2 + (y + 3)^2 = 2[/tex]
B. [tex](x - 4)^2 + (y - 3)^2 = 2[/tex]
C. [tex](x - 4)^2 + (y - 3)^2 = 2^2[/tex]
D. [tex](x + 4)^2 + (y + 3)^2 = 2^2[/tex]



Answer :

To determine the equation of the circle, we start by identifying the center and the radius. Let's start by analyzing the given equation of the circle:

[tex]\[ x^2 + y^2 - 8x - 6y + 24 = 0 \][/tex]

We need to rewrite this equation in the standard form of a circle's equation:

[tex]\[ (x-h)^2 + (y-k)^2 = r^2 \][/tex]

### Step-by-Step Solution:

1. Complete the square for the [tex]\(x\)[/tex] terms:

The portion involving [tex]\(x\)[/tex] is:

[tex]\[ x^2 - 8x \][/tex]

To complete the square:

[tex]\[ x^2 - 8x = (x - 4)^2 - 16 \][/tex]

2. Complete the square for the [tex]\(y\)[/tex] terms:

The portion involving [tex]\(y\)[/tex] is:

[tex]\[ y^2 - 6y \][/tex]

To complete the square:

[tex]\[ y^2 - 6y = (y - 3)^2 - 9 \][/tex]

3. Substitute back into the original equation:

Now we substitute into the original equation, replacing the terms with their completed square forms:

[tex]\[ (x - 4)^2 - 16 + (y - 3)^2 - 9 + 24 = 0 \][/tex]

4. Simplify the equation:

Combine constants:

[tex]\[ (x - 4)^2 + (y - 3)^2 - 25 + 24 = 0 \][/tex]

[tex]\[ (x - 4)^2 + (y - 3)^2 - 1 = 0 \][/tex]

Rearrange to the standard form:

[tex]\[ (x - 4)^2 + (y - 3)^2 = 1 \][/tex]

From this, we see that the center of the given circle is [tex]\( (4, 3) \)[/tex] and the radius is 1 unit.

5. Form the new equation:

We need to describe a new circle with the same center [tex]\( (4, 3) \)[/tex] but with a radius of 2 units (given in the problem).

The standard equation for a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:

[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]

For our circle,
- center [tex]\((4, 3)\)[/tex],
- radius [tex]\(2\)[/tex],

Thus, the equation becomes:

[tex]\[ (x - 4)^2 + (y - 3)^2 = 2^2 \][/tex]

This standard form equation of the new circle is:

[tex]\[ (x - 4)^2 + (y - 3)^2 = 4 \][/tex]

To match the provided choices:
[tex]\[ (x - 4)^2 + (y - 3)^2 = 2^2 \][/tex]

The correct equation is:

[tex]\[ \boxed{(x-4)^2+(y-3)^2=2^2} \][/tex]