To identify the correct equation of the circle that has its center at [tex]\((-16, 30)\)[/tex] and passes through the origin (0, 0), we need to follow these steps:
1. Identify the Center and the Radius of the Circle:
- The center of the circle is given by [tex]\((h, k) = (-16, 30)\)[/tex].
2. Find the Radius:
- The radius [tex]\(r\)[/tex] is the distance between the center [tex]\((-16, 30)\)[/tex] and the origin [tex]\((0, 0)\)[/tex].
- We find the radius using the distance formula:
[tex]\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
where [tex]\((x_1, y_1) = (-16, 30)\)[/tex] and [tex]\((x_2, y_2) = (0, 0)\)[/tex].
- Calculate:
[tex]\[
r = \sqrt{(-16 - 0)^2 + (30 - 0)^2} = \sqrt{(-16)^2 + (30)^2} = \sqrt{256 + 900} = \sqrt{1156}
\][/tex]
[tex]\[
r = 34
\][/tex]
3. Formulate the Equation of the Circle:
- The standard form of 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]
- Here, [tex]\(h = -16\)[/tex] and [tex]\(k = 30\)[/tex], and [tex]\(r = 34\)[/tex]. Thus,
[tex]\[
(x - (-16))^2 + (y - 30)^2 = 34^2
\][/tex]
- Simplify the equation:
[tex]\[
(x + 16)^2 + (y - 30)^2 = 34^2
\][/tex]
[tex]\[
(x + 16)^2 + (y - 30)^2 = 1156
\][/tex]
4. Verify the Correct Option:
- Now, we compare the formulated equation [tex]\((x + 16)^2 + (y - 30)^2 = 1156\)[/tex] with the given options:
[tex]\[
A. (x+16)^2 + (y-30)^2 = 34 \quad \text{(incorrect, wrong right side)}
\][/tex]
[tex]\[
B. (x-16)^2 + (y+30)^2 = 1156 \quad \text{(incorrect, wrong sign)}
\][/tex]
[tex]\[
C. (x-16)^2 + (y+30)^2 = 34 \quad \text{(incorrect, wrong signs and right side)}
\][/tex]
[tex]\[
D. (x+16)^2 + (y-30)^2 = 1156 \quad \text{(correct, matches our formulated equation)}
\][/tex]
Therefore, the correct equation of the circle is given by option D:
[tex]\[
(x+16)^2 + (y-30)^2 = 1156
\][/tex]
