To determine which point lies on the circle represented by the equation [tex]\((x+7)^2 + (y-10)^2 = 13^2\)[/tex], we will plug each point into the circle's equation and check if the equation holds true.
### Analyzing Option A: [tex]\((5, 12)\)[/tex]
Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = 12 \)[/tex] into the equation:
[tex]\[
(5+7)^2 + (12-10)^2 = 13^2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
(12)^2 + (2)^2 = 169
\][/tex]
Now, calculate the squares:
[tex]\[
144 + 4 = 148
\][/tex]
Since [tex]\( 148 \neq 169 \)[/tex], point [tex]\((5, 12)\)[/tex] does not lie on the circle.
### Analyzing Option B: [tex]\((-7, -3)\)[/tex]
Substitute [tex]\( x = -7 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation:
[tex]\[
(-7+7)^2 + (-3-10)^2 = 13^2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
(0)^2 + (-13)^2 = 169
\][/tex]
Now, calculate the squares:
[tex]\[
0 + 169 = 169
\][/tex]
Since [tex]\( 169 = 169 \)[/tex], point [tex]\((-7, -3)\)[/tex] lies on the circle.
### Analyzing Option C: [tex]\((-6, -10)\)[/tex]
Substitute [tex]\( x = -6 \)[/tex] and [tex]\( y = -10 \)[/tex] into the equation:
[tex]\[
(-6+7)^2 + (-10-10)^2 = 13^2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
(1)^2 + (-20)^2 = 169
\][/tex]
Now, calculate the squares:
[tex]\[
1 + 400 = 401
\][/tex]
Since [tex]\( 401 \neq 169 \)[/tex], point [tex]\((-6, -10)\)[/tex] does not lie on the circle.
### Analyzing Option D: [tex]\((6, 23)\)[/tex]
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = 23 \)[/tex] into the equation:
[tex]\[
(6+7)^2 + (23-10)^2 = 13^2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
(13)^2 + (13)^2 = 169
\][/tex]
Now, calculate the squares:
[tex]\[
169 + 169 = 338
\][/tex]
Since [tex]\( 338 \neq 169 \)[/tex], point [tex]\((6, 23)\)[/tex] does not lie on the circle.
### Conclusion:
The point that lies on the circle represented by the equation [tex]\((x+7)^2 + (y-10)^2 = 13^2\)[/tex] is:
B. [tex]\((-7, -3)\)[/tex]
