To determine which of the given coordinates lie on the circle defined by the equation:
[tex]\[
(x-3)^2 + y^2 + 8y = 84
\][/tex]
we first need to convert this equation into the standard form of a circle equation, which 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. Here are the steps to convert and analyze the equation:
1. Original Equation: [tex]\((x-3)^2 + y^2 + 8y = 84\)[/tex]
2. Rearrange the [tex]\(y\)[/tex] Terms: We need to complete the square for the terms involving [tex]\(y\)[/tex]. Consider the expression [tex]\(y^2 + 8y\)[/tex]:
- Add and subtract [tex]\((4)^2 = 16\)[/tex] to complete the square:
[tex]\[
y^2 + 8y + 16 - 16
\][/tex]
3. Rewrite the Equation:
[tex]\[
(x-3)^2 + (y^2 + 8y + 16 - 16) = 84
\][/tex]
[tex]\[
(x-3)^2 + (y + 4)^2 - 16 = 84
\][/tex]
4. Simplify:
[tex]\[
(x-3)^2 + (y+4)^2 - 16 = 84
\][/tex]
Add 16 to both sides:
[tex]\[
(x-3)^2 + (y+4)^2 = 100
\][/tex]
This transformed equation [tex]\((x-3)^2 + (y+4)^2 = 100\)[/tex] represents a circle with center [tex]\((3, -4)\)[/tex] and radius [tex]\(10\)[/tex] (since [tex]\(100 = 10^2\)[/tex]).
Next, we need to check which of the coordinates lie on this circle by substituting the coordinates into the equation and seeing if they satisfy it. Let’s check each of the provided points:
Point A: (1, 7)
[tex]\[
(x-3)^2 + (y+4)^2 = (1-3)^2 + (7+4)^2 = (-2)^2 + (11)^2 = 4 + 121 = 125
\][/tex]
[tex]\(125 \neq 100\)[/tex], so point (1, 7) does not lie on the circle.
Point B: (-2, 5)
[tex]\[
(x-3)^2 + (y+4)^2 = (-2-3)^2 + (5+4)^2 = (-5)^2 + (9)^2 = 25 + 81 = 106
\][/tex]
[tex]\(106 \neq 100\)[/tex], so point (-2, 5) does not lie on the circle.
Point C: (-3, 4)
[tex]\[
(x-3)^2 + (y+4)^2 = (-3-3)^2 + (4+4)^2 = (-6)^2 + (8)^2 = 36 + 64 = 100
\][/tex]
[tex]\(100 = 100\)[/tex], so point (-3, 4) does lie on the circle.
Point D: (3, -6)
[tex]\[
(x-3)^2 + (y+4)^2 = (3-3)^2 + (-6+4)^2 = (0)^2 + (-2)^2 = 0 + 4 = 4
\][/tex]
[tex]\(4 \neq 100\)[/tex], so point (3, -6) does not lie on the circle.
Thus, the only coordinate that lies on the circle is:
[tex]\[
\boxed{(-3,4)}
\][/tex]
