To find which equation represents a circle that contains the point \((-5,-3)\) and has a center at \((-2,1)\), we'll follow these steps:
1. Calculate the Radius:
Using the distance formula, we calculate the radius of the circle. The formula for the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[
d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}
\][/tex]
Substituting the given points:
[tex]\[
d = \sqrt{\left(-5 - (-2)\right)^2 + \left(-3 - 1\right)^2}
\][/tex]
[tex]\[
d = \sqrt{\left(-3\right)^2 + \left(-4\right)^2}
\][/tex]
[tex]\[
d = \sqrt{9 + 16}
\][/tex]
[tex]\[
d = \sqrt{25}
\][/tex]
[tex]\[
d = 5
\][/tex]
So, the radius \(r\) of the circle is 5.
2. Check Given Equations:
Now, let's identify which of the given equations describe a circle with radius 5 centered at \((-2,1)\).
- Equation 1: \((x - 1)^2 + (y + 2)^2 = 25\)
[tex]\[
\text{Center} = (1, -2), \quad \text{Radius} = \sqrt{25} = 5
\][/tex]
This equation has the correct radius, but the center is incorrect.
- Equation 2: \((x + 2)^2 + (y - 1)^2 = 5\)
[tex]\[
\text{Center} = (-2, 1), \quad \text{Radius} = \sqrt{5} \neq 5
\][/tex]
This equation has the correct center, but the radius is incorrect.
- Equation 3: \((x + 2)^2 + (y - 1)^2 = 25\)
[tex]\[
\text{Center} = (-2, 1), \quad \text{Radius} = \sqrt{25} = 5
\][/tex]
This equation has both the correct center and radius.
- Equation 4: \((x - 1)^2 + (y + 2)^2 = 5\)
[tex]\[
\text{Center} = (1, -2), \quad \text{Radius} = \sqrt{5} \neq 5
\][/tex]
This equation has incorrect center and radius.
Given that Equation 3 has both the correct center \((-2, 1)\) and the correct radius \(r = 5\):
The equation that represents a circle which contains the point \((-5,-3)\) and has a center at \((-2,1)\) is:
[tex]\[(x + 2)^2 + (y - 1)^2 = 25\][/tex]
