To identify the equation of the circle that has its center at [tex]\((-8, 15)\)[/tex] and passes through the origin, we follow these steps:
1. Determine the center of the circle:
The center is given as [tex]\((-8, 15)\)[/tex].
2. Find the radius of the circle:
The radius can be determined using the distance formula between the center of the circle [tex]\((-8, 15)\)[/tex] and the origin [tex]\((0, 0)\)[/tex].
The distance formula is:
[tex]\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the coordinates of the center and the origin:
[tex]\[
r = \sqrt{(-8 - 0)^2 + (15 - 0)^2}
\][/tex]
Simplifying within the square root:
[tex]\[
r = \sqrt{(-8)^2 + (15)^2} = \sqrt{64 + 225} = \sqrt{289}
\][/tex]
Therefore, the radius [tex]\( r \)[/tex] is:
[tex]\[
r = 17
\][/tex]
3. Write the general equation of the circle:
The general 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]
Given the center [tex]\((-8, 15)\)[/tex] and radius [tex]\( 17 \)[/tex], we substitute [tex]\(h = -8\)[/tex], [tex]\( k = 15\)[/tex], and [tex]\( r = 17 \)[/tex]:
[tex]\[
(x - (-8))^2 + (y - 15)^2 = 17^2
\][/tex]
Simplifying the expression, this becomes:
[tex]\[
(x + 8)^2 + (y - 15)^2 = 289
\][/tex]
So the equation of the circle is:
[tex]\[
(x + 8)^2 + (y - 15)^2 = 289
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{A. \ (x+8)^2+(y-15)^2=289}
\][/tex]
