To determine which point lies on the circle represented by the equation [tex]\( x^2 + (y - 12)^2 = 25^2 \)[/tex], we need to check if each point satisfies the circle's equation.
Given points:
- [tex]\( (20, -3) \)[/tex]
- [tex]\( (-7, 24) \)[/tex]
- [tex]\( (0, 13) \)[/tex]
- [tex]\( (-25, -13) \)[/tex]
Substituting these points into the equation [tex]\( x^2 + (y - 12)^2 = 625 \)[/tex], we get:
1. For the point [tex]\( (20, -3) \)[/tex]:
[tex]\[
x = 20, \quad y = -3
\][/tex]
[tex]\[
20^2 + (-3 - 12)^2 = 400 + (-15)^2 = 400 + 225 = 625
\][/tex]
[tex]\[
625 = 625
\][/tex]
[tex]\( (20, -3) \)[/tex] satisfies the equation of the circle.
2. For the point [tex]\( (-7, 24) \)[/tex]:
[tex]\[
x = -7, \quad y = 24
\][/tex]
[tex]\[
(-7)^2 + (24 - 12)^2 = 49 + 12^2 = 49 + 144 = 193
\][/tex]
[tex]\[
193 \neq 625
\][/tex]
[tex]\( (-7, 24) \)[/tex] does not satisfy the equation of the circle.
3. For the point [tex]\( (0, 13) \)[/tex]:
[tex]\[
x = 0, \quad y = 13
\][/tex]
[tex]\[
0^2 + (13 - 12)^2 = 0 + 1 = 1
\][/tex]
[tex]\[
1 \neq 625
\][/tex]
[tex]\( (0, 13) \)[/tex] does not satisfy the equation of the circle.
4. For the point [tex]\( (-25, -13) \)[/tex]:
[tex]\[
x = -25, \quad y = -13
\][/tex]
[tex]\[
(-25)^2 + (-13 - 12)^2 = 625 + (-25)^2 = 625 + 625 = 1250
\][/tex]
[tex]\[
1250 \neq 625
\][/tex]
[tex]\( (-25, -13) \)[/tex] does not satisfy the equation of the circle.
Conclusively, the only point that lies on the circle [tex]\( x^2 + (y - 12)^2 = 625 \)[/tex] is [tex]\( (20, -3) \)[/tex].
Thus, the correct answer is:
A. [tex]\( (20, -3) \)[/tex]
