The question asks us to determine which point lies on the circle represented by the equation [tex]\( x^2 + (y - 12)^2 = 25^2 \)[/tex].
Let's break this down step-by-step:
### Circle Equation
The given circle equation is:
[tex]\[ x^2 + (y - 12)^2 = 25^2 \][/tex]
This represents a circle with:
- Center at [tex]\((0, 12)\)[/tex]
- Radius of 25
### Checking each point
We need to check each of the given points to see which one satisfies the circle's equation.
#### Point A: [tex]\((20, -3)\)[/tex]
Substitute [tex]\(x = 20\)[/tex] and [tex]\(y = -3\)[/tex] into the circle's equation:
[tex]\[ 20^2 + (-3 - 12)^2 = 400 + (-15)^2 = 400 + 225 = 625 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
This is a true statement. So, point [tex]\((20, -3)\)[/tex] lies on the circle.
#### Point B: [tex]\((-7, 24)\)[/tex]
Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = 24\)[/tex] into the circle's equation:
[tex]\[ (-7)^2 + (24 - 12)^2 = 49 + 12^2 = 49 + 144 = 193 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
This is false. So, point [tex]\((-7, 24)\)[/tex] does not lie on the circle.
#### Point C: [tex]\((0, 13)\)[/tex]
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 13\)[/tex] into the circle's equation:
[tex]\[ 0^2 + (13 - 12)^2 = 0 + 1^2 = 1 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
This is false. So, point [tex]\((0, 13)\)[/tex] does not lie on the circle.
#### Point D: [tex]\((-25, -13)\)[/tex]
Substitute [tex]\(x = -25\)[/tex] and [tex]\(y = -13\)[/tex] into the circle's equation:
[tex]\[ (-25)^2 + (-13 - 12)^2 = 625 + (-25)^2 = 625 + 625 = 1250 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
This is false. So, point [tex]\((-25, -13)\)[/tex] does not lie on the circle.
### Conclusion
After checking all the points, we find that the only point that satisfies the circle's equation is:
- Point A [tex]\((20, -3)\)[/tex]
Therefore, the correct answer is:
A. [tex]\( (20, -3) \)[/tex]
