Question
A circle is drawn on a coordinate plane with the center point at (−1,3) and a radius of 5. Derive the equation of the circle in standard form, using the given information and the Pythagorean Theorem.(1 point)
Responses
(x−3)2+(y−1)2=25
left parenthesis x minus 3 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 25
(x+3)2+(y−1)2=25
left parenthesis x plus 3 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 25
(x−1)2+(y−3)2=25
left parenthesis x minus 1 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 25
(x+1)2+(y−3)2=25
Responses
(x−3)2+(y−1)2=25
left parenthesis x minus 3 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 25
(x+3)2+(y−1)2=25
left parenthesis x plus 3 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 25
(x−1)2+(y−3)2=25
left parenthesis x minus 1 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 25
(x+1)2+(y−3)2=25
Answers
GPT-4o mini
The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by the formula:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the center of the circle is at \((-1, 3)\) and the radius is \(5\).
1. Substitute \(h = -1\), \(k = 3\), and \(r = 5\) into the formula:
\[
(x - (-1))^2 + (y - 3)^2 = 5^2
\]
2. Simplify the equation:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Thus, the correct equation of the circle in standard form is:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
So the correct response from the options provided is:
\((x + 1)^2 + (y - 3)^2 = 25\).
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the center of the circle is at \((-1, 3)\) and the radius is \(5\).
1. Substitute \(h = -1\), \(k = 3\), and \(r = 5\) into the formula:
\[
(x - (-1))^2 + (y - 3)^2 = 5^2
\]
2. Simplify the equation:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Thus, the correct equation of the circle in standard form is:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
So the correct response from the options provided is:
\((x + 1)^2 + (y - 3)^2 = 25\).