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 this equation:
1. Point A: [tex]\((20, -3)\)[/tex]
- Substitute [tex]\(x = 20\)[/tex] and [tex]\(y = -3\)[/tex] into the circle equation:
[tex]\[
20^2 + (-3 - 12)^2 = 25^2
\][/tex]
Simplifying inside the parentheses:
[tex]\[
20^2 + (-15)^2 = 25^2
\][/tex]
Squaring the values:
[tex]\[
400 + 225 = 625
\][/tex]
Since [tex]\(400 + 225 = 625\)[/tex], point A satisfies the equation.
2. Point B: [tex]\((-7, 24)\)[/tex]
- Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = 24\)[/tex] into the circle equation:
[tex]\[
(-7)^2 + (24 - 12)^2 = 25^2
\][/tex]
Simplifying inside the parentheses:
[tex]\[
49 + 12^2 = 25^2
\][/tex]
Squaring the values:
[tex]\[
49 + 144 = 625
\][/tex]
Since [tex]\(49 + 144 = 193\)[/tex], point B does not satisfy the equation.
3. Point C: [tex]\((0, 13)\)[/tex]
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 13\)[/tex] into the circle equation:
[tex]\[
0^2 + (13 - 12)^2 = 25^2
\][/tex]
Simplifying inside the parentheses:
[tex]\[
0 + 1^2 = 25^2
\][/tex]
Squaring the values:
[tex]\[
0 + 1 = 625
\][/tex]
Since [tex]\(1 \neq 625\)[/tex], point C does not satisfy the equation.
4. Point D: [tex]\((-25, -13)\)[/tex]
- Substitute [tex]\(x = -25\)[/tex] and [tex]\(y = -13\)[/tex] into the circle equation:
[tex]\[
(-25)^2 + (-13 - 12)^2 = 25^2
\][/tex]
Simplifying inside the parentheses:
[tex]\[
625 + (-25)^2 = 25^2
\][/tex]
Squaring the values:
[tex]\[
625 + 625 = 625
\][/tex]
Since [tex]\(1250 \neq 625\)[/tex], point D does not satisfy the equation.
Given the calculations, we can confirm that point [tex]\((20, -3)\)[/tex] lies on the circle.
So, the correct answer is:
A. [tex]\((20,-3)\)[/tex]
