To determine which point lies on the circle represented by the equation [tex]\( x^2 + (y-12)^2 = 25^2 \)[/tex], we need to substitute each point into the equation and check if it holds true.
The equation of the circle is given by:
[tex]\[ x^2 + (y-12)^2 = 25^2 \][/tex]
which simplifies to:
[tex]\[ x^2 + (y-12)^2 = 625 \][/tex]
Let's verify each point:
Point A: [tex]\((20, -3)\)[/tex]
Substitute [tex]\( x = 20 \)[/tex] and [tex]\( y = -3 \)[/tex]:
[tex]\[ 20^2 + (-3 - 12)^2 = 20^2 + (-15)^2 = 400 + 225 = 625 \][/tex]
The left-hand side equals the right-hand side of the equation [tex]\(625 = 625\)[/tex].
Point B: [tex]\((-7, 24)\)[/tex]
Substitute [tex]\( x = -7 \)[/tex] and [tex]\( y = 24 \)[/tex]:
[tex]\[ (-7)^2 + (24 - 12)^2 = 49 + 12^2 = 49 + 144 = 193 \][/tex]
The left-hand side does not equal the right-hand side [tex]\(193 \neq 625\)[/tex].
Point C: [tex]\((0, 13)\)[/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 13 \)[/tex]:
[tex]\[ 0^2 + (13 - 12)^2 = 0 + 1 = 1 \][/tex]
The left-hand side does not equal the right-hand side [tex]\(1 \neq 625\)[/tex].
Point D: [tex]\((-25, -13)\)[/tex]
Substitute [tex]\( x = -25 \)[/tex] and [tex]\( y = -13 \)[/tex]:
[tex]\[ (-25)^2 + (-13 - 12)^2 = 625 + (-25)^2 = 625 + 625 = 1250 \][/tex]
The left-hand side does not equal the right-hand side [tex]\(1250 \neq 625\)[/tex].
According to these calculations, only point A (20, -3) lies on the circle.
Therefore, the correct answer is:
A. [tex]\((20, -3)\)[/tex]
