Question
A city plans to build a circular park around a historical site. The center of this park is located at (3,−2) . If the park's radius is 7 units, representing the boundary in meters, what is the equation of the circular boundary?(1 point) Responses x2+y2=49 x squared plus y squared is equal to 49 (x+3)2+(y−2)2=7 open paren x plus 3 close paren squared plus open paren y minus 2 close paren squared is equal to 7 (x−3)2+(y+2)2=49 open paren x minus 3 close paren squared plus open paren y plus 2 close paren squared is equal to 49 (x−3)2+(y+2)2=7
Answers
GPT-4o mini
The standard equation of a circle with its center at point \((h, k)\) and a radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the center of the park is located at \((3, -2)\) and the radius is \(7\) units. Therefore, we have:
- \(h = 3\)
- \(k = -2\)
- \(r = 7\)
Substituting these values into the equation, we get:
\[
(x - 3)^2 + (y + 2)^2 = 7^2
\]
Which simplifies to:
\[
(x - 3)^2 + (y + 2)^2 = 49
\]
Thus, the correct equation of the circular boundary is:
\[
\boxed{(x - 3)^2 + (y + 2)^2 = 49}
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the center of the park is located at \((3, -2)\) and the radius is \(7\) units. Therefore, we have:
- \(h = 3\)
- \(k = -2\)
- \(r = 7\)
Substituting these values into the equation, we get:
\[
(x - 3)^2 + (y + 2)^2 = 7^2
\]
Which simplifies to:
\[
(x - 3)^2 + (y + 2)^2 = 49
\]
Thus, the correct equation of the circular boundary is:
\[
\boxed{(x - 3)^2 + (y + 2)^2 = 49}
\]