Which equation represents a circle that contains the point \((-5,-3)\) and has a center at \((-2,1)\)?

Distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

A. \((x - 1)^2 + (y + 2)^2 = 25\)
B. \((x + 2)^2 + (y - 1)^2 = 5\)
C. \((x + 2)^2 + (y - 1)^2 = 25\)
D. [tex]\((x - 1)^2 + (y + 2)^2 = 5\)[/tex]



Answer :

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]