Answer :
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]
### 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]