To determine which equation represents a circle that contains the point [tex]\((-5, -3)\)[/tex] and has a center at [tex]\((-2,1)\)[/tex], we need to follow a detailed step-by-step approach:
### Step 1: Calculate the radius of the circle
The radius [tex]\(r\)[/tex] of the circle can be found using the distance formula between the point [tex]\((-5, -3)\)[/tex] and the center [tex]\((-2, 1)\)[/tex]:
[tex]\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substitute the given coordinates:
[tex]\[
r = \sqrt{((-2) - (-5))^2 + ((1) - (-3))^2}
\][/tex]
Simplify the expressions within the square root:
[tex]\[
= \sqrt{(-2 + 5)^2 + (1 + 3)^2}
= \sqrt{3^2 + 4^2}
= \sqrt{9 + 16}
= \sqrt{25}
= 5
\][/tex]
Thus, the radius [tex]\(r\)[/tex] is 5.
### Step 2: Write the equation of the circle
The standard equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is given by:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
For our circle, the center is [tex]\((-2, 1)\)[/tex] and the radius [tex]\(r\)[/tex] is 5. Hence, [tex]\(h = -2\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(r^2 = 25\)[/tex]. Therefore, the equation of our circle is:
[tex]\[
(x + 2)^2 + (y - 1)^2 = 25
\][/tex]
### Step 3: Identify the correct equation from the given options
Given the options are:
1. [tex]\((x - 1)^2 + (y + 2)^2 = 25\)[/tex]
2. [tex]\((x + 2)^2 + (y - 1)^2 = 5\)[/tex]
3. [tex]\((x + 2)^2 + (y - 1)^2 = 25\)[/tex]
4. [tex]\((x - 1)^2 + (y + 2)^2 = 5\)[/tex]
The correct equation from the options is:
[tex]\(
\boxed{(x + 2)^2 + (y - 1)^2 = 25}
\)[/tex]
