To identify the equation of the circle that has its center at [tex]\((-16,30)\)[/tex] and passes through the origin [tex]\((0, 0)\)[/tex], we need to follow these steps:
1. Determine the radius of the circle:
- The radius [tex]\( r \)[/tex] of the circle can be found by calculating the distance from the center to the point through which it passes (the origin in this case).
- The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Here, the coordinates of the center are [tex]\((x_1, y_1) = (-16, 30)\)[/tex], and the coordinates of the origin are [tex]\((x_2, y_2) = (0, 0)\)[/tex].
[tex]\[
r = \sqrt{(0 - (-16))^2 + (0 - 30)^2}
\][/tex]
[tex]\[
r = \sqrt{(16)^2 + (-30)^2}
\][/tex]
[tex]\[
r = \sqrt{256 + 900}
\][/tex]
[tex]\[
r = \sqrt{1156}
\][/tex]
[tex]\[
r = 34
\][/tex]
2. Write the general 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]
- In our case, the center [tex]\((h, k) = (-16, 30)\)[/tex] and the radius [tex]\(r = 34\)[/tex].
- So, plugging in these values, the equation becomes:
[tex]\[
(x - (-16))^2 + (y - 30)^2 = 34^2
\][/tex]
[tex]\[
(x + 16)^2 + (y - 30)^2 = 1156
\][/tex]
3. Compare with the options provided:
- The given options are:
- 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 form of the equation we derived is:
[tex]\[
(x + 16)^2 + (y - 30)^2 = 1156
\][/tex]
- Comparing this with the options, we see that option D matches perfectly.
Therefore, the equation of the circle that has its center at [tex]\((-16,30)\)[/tex] and passes through the origin is:
[tex]\[
\boxed{(x + 16)^2 + (y - 30)^2 = 1156}
\][/tex]
