Let's solve this step-by-step to determine which point lies on the circle described by the equation:
[tex]\[ x^2 + (y - 12)^2 = 25^2 \][/tex]
First, let's rewrite the circle's equation for clarity:
[tex]\[ x^2 + (y - 12)^2 = 625 \][/tex]
We need to check each given point to see if it satisfies this equation.
Point A: [tex]\((20, -3)\)[/tex]
Substitute [tex]\(x = 20\)[/tex] and [tex]\(y = -3\)[/tex] into the circle's equation:
[tex]\[ 20^2 + (-3 - 12)^2 = 625 \][/tex]
[tex]\[ 400 + (-15)^2 = 625 \][/tex]
[tex]\[ 400 + 225 = 625 \][/tex]
[tex]\[ 625 = 625 \][/tex]
This point satisfies the equation, so point A lies on the circle.
Point B: [tex]\((-7, 24)\)[/tex]
Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = 24\)[/tex]:
[tex]\[ (-7)^2 + (24 - 12)^2 = 625 \][/tex]
[tex]\[ 49 + (12)^2 = 625 \][/tex]
[tex]\[ 49 + 144 = 625 \][/tex]
[tex]\[ 193 \neq 625 \][/tex]
This point does not satisfy the equation, so point B does not lie on the circle.
Point C: [tex]\((0, 13)\)[/tex]
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 13\)[/tex]:
[tex]\[ 0^2 + (13 - 12)^2 = 625 \][/tex]
[tex]\[ 0 + 1 = 625 \][/tex]
[tex]\[ 1 \neq 625 \][/tex]
This point does not satisfy the equation, so point C does not lie on the circle.
Point D: [tex]\((-25, -13)\)[/tex]
Substitute [tex]\(x = -25\)[/tex] and [tex]\(y = -13\)[/tex]:
[tex]\[ (-25)^2 + (-13 - 12)^2 = 625 \][/tex]
[tex]\[ 625 + (-25)^2 = 625 \][/tex]
[tex]\[ 625 + 625 = 625 \][/tex]
[tex]\[ 1250 \neq 625 \][/tex]
This point does not satisfy the equation, so point D does not lie on the circle.
After checking all points, we find that the point which lies on the circle is:
[tex]\[ \boxed{A \; (20, -3)} \][/tex]
