To identify the correct equation of the circle that has its center at [tex]\((7, -24)\)[/tex] and passes through the origin, we will follow these steps:
1. Identify the Center of the Circle:
The center of the circle is given as [tex]\((7, -24)\)[/tex]. This gives us the coordinates [tex]\((h, k) = (7, -24)\)[/tex].
2. Determine the Radius:
The radius [tex]\( r \)[/tex] of the circle can be found by calculating the distance from the center of the circle to the origin [tex]\((0, 0)\)[/tex]. We use the distance formula for this calculation:
[tex]\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the values of the coordinates of the center and the origin into the distance formula, we get:
[tex]\[
r = \sqrt{(7 - 0)^2 + (-24 - 0)^2} = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25
\][/tex]
3. Form the Equation of the Circle:
The general equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is given by:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting the values [tex]\( h = 7 \)[/tex], [tex]\( k = -24 \)[/tex], and [tex]\( r = 25 \)[/tex], we get:
[tex]\[
(x - 7)^2 + (y + 24)^2 = 25^2
\][/tex]
Since [tex]\( r^2 = 25^2 = 625 \)[/tex], the equation simplifies to:
[tex]\[
(x - 7)^2 + (y + 24)^2 = 625
\][/tex]
4. Match the Equation with the Given Choices:
Look at the provided choices to find the one that matches our derived equation:
- A. [tex]\((x+7)^2 + (y-24)^2 = 25\)[/tex]
- B. [tex]\((x-7)^2 + (y+24)^2 = 25\)[/tex]
- C. [tex]\((x+7)^2 + (y-24)^2 = 625\)[/tex]
- D. [tex]\((x-7)^2 + (y+24)^2 = 625\)[/tex]
Comparing this with our equation [tex]\((x - 7)^2 + (y + 24)^2 = 625\)[/tex], we see that the correct choice is:
D. [tex]\((x-7)^2 + (y+24)^2 = 625\)[/tex]
Thus, the equation of the circle that has its center at [tex]\((7, -24)\)[/tex] and passes through the origin is:
[tex]\[
\boxed{(x-7)^2 + (y+24)^2 = 625}
\][/tex]
