To find the equation of a circle with a given center and passing through a specific point, we follow these steps:
1. Identify the Center of the Circle: The center is given as [tex]\((h, k) = (-16, 30)\)[/tex].
2. Determine the Radius: We need to find the distance from the center to any point on the circle. Let's choose a point directly on the circle as [tex]\((x_1, y_1)\)[/tex]. If the circle passes through a point directly below the center by 20 units, our point would be at [tex]\((-16, 30 - 20) = (-16, 10)\)[/tex].
3. Distance Formula: The radius [tex]\(r\)[/tex] is the distance between the center [tex]\((-16, 30)\)[/tex] and the point [tex]\((-16, 10)\)[/tex]. We use the distance formula:
[tex]\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the points:
[tex]\[
r = \sqrt{(-16 - (-16))^2 + (10 - 30)^2} = \sqrt{0 + (-20)^2} = \sqrt{400} = 20
\][/tex]
4. Radius Squared: From the calculation, the radius [tex]\(r = 20\)[/tex], hence [tex]\(r^2 = 400\)[/tex].
5. Standard Form of Circle's Equation: The equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting [tex]\(h = -16\)[/tex], [tex]\(k = 30\)[/tex], and [tex]\(r^2 = 400\)[/tex]:
[tex]\[
(x + 16)^2 + (y - 30)^2 = 400
\][/tex]
Comparing the obtained equation with the options provided:
A. [tex]\((x - 16)^2 + (y + 30)^2 = 1256\)[/tex]
B. [tex]\((x + 16)^2 + (y - 30)^2 = 1256\)[/tex]
C. [tex]\((x - 16)^2 + (y + 30)^2 = 1156\)[/tex]
D. [tex]\((x + 16)^2 + (y - 30)^2 = 1156\)[/tex]
The correct equation is:
[tex]\[
D: (x + 16)^2 + (y - 30)^2 = 400
\][/tex]
None of the given options exactly match this equation.
Hence if we have to choose option between that are the closest to our result, is:
[tex]\[
B: (x + 16)^2 + (y - 30)^2 which only differs in the radius squared
\][/tex]
