To find the equation of a circle with a center at [tex]\((-4,0)\)[/tex] that passes through the point [tex]\((-2,1)\)[/tex], we will follow the steps outlined.
### Step 1: Use the Distance Formula to Determine the Radius
The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\((x_1, y_1) = (-4, 0)\)[/tex] is the center of the circle, and [tex]\((x_2, y_2) = (-2, 1)\)[/tex] is a point on the circle. Plugging in these values:
[tex]\[ \text{Radius} = \sqrt{((-2) - (-4))^2 + (1 - 0)^2} \][/tex]
[tex]\[ = \sqrt{(-2 + 4)^2 + 1^2} \][/tex]
[tex]\[ = \sqrt{(2)^2 + 1^2} \][/tex]
[tex]\[ = \sqrt{4 + 1} \][/tex]
[tex]\[ = \sqrt{5} \][/tex]
[tex]\[ \approx 2.236 \][/tex]
Since the radius is [tex]\(\sqrt{5}\)[/tex], the radius squared ([tex]\(r^2\)[/tex]) is:
[tex]\[ r^2 = (\sqrt{5})^2 = 5 \][/tex]
### Step 2: Substitute Known Values into the Standard Form
The standard form of the equation of a circle is given by:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius. Given the center at [tex]\((-4,0)\)[/tex] and [tex]\(r^2 = 5\)[/tex], we substitute [tex]\(h = -4\)[/tex], [tex]\(k = 0\)[/tex], and [tex]\(r^2 = 5\)[/tex]:
[tex]\[ (x - (-4))^2 + (y - 0)^2 = 5 \][/tex]
[tex]\[ (x + 4)^2 + y^2 = 5 \][/tex]
Therefore, the equation of the circle is:
[tex]\[ (x + 4)^2 + y^2 = 5 \][/tex]
None of the suggested options matches exactly. Hence, the correct equation of the circle with a center at [tex]\((-4,0)\)[/tex] that passes through the point [tex]\((-2,1)\)[/tex] is:
[tex]\[ (x + 4)^2 + y^2 = 5 \][/tex]
