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 each given point against the circle’s equation.
The general form of a circle's equation is [tex]\( (x-h)^2 + (y-k)^2 = r^2 \)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle, and [tex]\( r \)[/tex] is the radius.
For this specific equation:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((0, 12)\)[/tex].
- The radius [tex]\( r \)[/tex] is [tex]\( 25 \)[/tex].
We will check each point.
### Point A: [tex]\((20, -3)\)[/tex]
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[ 20^2 = 400 \][/tex]
2. Calculate [tex]\( (y - 12)^2 \)[/tex]:
[tex]\[ (-3 - 12)^2 = (-15)^2 = 225 \][/tex]
3. Add these together:
[tex]\[ 400 + 225 = 625 \][/tex]
Since [tex]\( 625 = 25^2 \)[/tex], point A [tex]\((20, -3)\)[/tex] lies on the circle.
### Point B: [tex]\((-7, 24)\)[/tex]
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[ (-7)^2 = 49 \][/tex]
2. Calculate [tex]\( (y - 12)^2 \)[/tex]:
[tex]\[ (24 - 12)^2 = 12^2 = 144 \][/tex]
3. Add these together:
[tex]\[ 49 + 144 = 193 \][/tex]
Since [tex]\( 193 \neq 625 \)[/tex], point B [tex]\((-7, 24)\)[/tex] does not lie on the circle.
### Point C: [tex]\((0, 13)\)[/tex]
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[ 0^2 = 0 \][/tex]
2. Calculate [tex]\( (y - 12)^2 \)[/tex]:
[tex]\[ (13 - 12)^2 = 1^2 = 1 \][/tex]
3. Add these together:
[tex]\[ 0 + 1 = 1 \][/tex]
Since [tex]\( 1 \neq 625 \)[/tex], point C [tex]\((0, 13)\)[/tex] does not lie on the circle.
### Point D: [tex]\((-25, -13)\)[/tex]
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[ (-25)^2 = 625 \][/tex]
2. Calculate [tex]\( (y - 12)^2 \)[/tex]:
[tex]\[ (-13 - 12)^2 = (-25)^2 = 625 \][/tex]
3. Add these together:
[tex]\[ 625 + 625 = 1250 \][/tex]
Since [tex]\( 1250 \neq 625 \)[/tex], point D [tex]\((-25, -13)\)[/tex] does not lie on the circle.
Thus, the correct answer is:
A. [tex]\((20,-3)\)[/tex]
