To determine which point lies on the circle given by the equation [tex]\((x+7)^2 + (y-10)^2 = 13^2\)[/tex], we need to check the distance of each point from the center of the circle and see if it equals the radius of the circle.
The center of the circle is [tex]\((-7, 10)\)[/tex] and the radius is 13.
We will calculate the distance of each point from the center of the circle:
1. For point [tex]\((5, 12)\)[/tex]:
[tex]\[
\sqrt{(5 + 7)^2 + (12 - 10)^2} = \sqrt{12^2 + 2^2} = \sqrt{144 + 4} = \sqrt{148} \approx 12.17
\][/tex]
2. For point [tex]\((-7, -3)\)[/tex]:
[tex]\[
\sqrt{(-7 + 7)^2 + (-3 - 10)^2} = \sqrt{0^2 + (-13)^2} = \sqrt{169} = 13
\][/tex]
3. For point [tex]\((-6, -10)\)[/tex]:
[tex]\[
\sqrt{(-6 + 7)^2 + (-10 - 10)^2} = \sqrt{1^2 + (-20)^2} = \sqrt{1 + 400} = \sqrt{401} \approx 20.02
\][/tex]
4. For point [tex]\((6, 23)\)[/tex]:
[tex]\[
\sqrt{(6 + 7)^2 + (23 - 10)^2} = \sqrt{13^2 + 13^2} = \sqrt{169 + 169} = \sqrt{338} \approx 18.38
\][/tex]
Comparing the distances calculated:
- Distance for [tex]\((5, 12)\)[/tex] is approximately [tex]\(12.17\)[/tex], not equal to 13.
- Distance for [tex]\((-7, -3)\)[/tex] is exactly [tex]\(13\)[/tex], which is equal to 13.
- Distance for [tex]\((-6, -10)\)[/tex] is approximately [tex]\(20.02\)[/tex], not equal to 13.
- Distance for [tex]\((6, 23)\)[/tex] is approximately [tex]\(18.38\)[/tex], not equal to 13.
Therefore, the point that lies on the circle [tex]\((x+7)^2 + (y-10)^2 = 13^2\)[/tex] is [tex]\((-7, -3)\)[/tex].
The correct answer is:
B. [tex]\((-7, -3)\)[/tex]
