Question
Which equation is the standard form of the equation of a circle drawn on the coordinate plane, derived from the Pythagorean theorem? The radius is 5 and the center point is (0,0).
X^2+y^2=25
(X-4)^2+(y-3)^2=25
X^2+(y-5)^2=25
(X+4)^2+(y+3)^2=25
X^2+y^2=25
(X-4)^2+(y-3)^2=25
X^2+(y-5)^2=25
(X+4)^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:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Given that the center of the circle is \((0,0)\) and the radius is \(5\), we can substitute \(h = 0\), \(k = 0\), and \(r = 5\) into the formula:
\[
(x - 0)^2 + (y - 0)^2 = 5^2
\]
This simplifies to:
\[
x^2 + y^2 = 25
\]
Therefore, the correct equation of the circle in standard form is:
**\(x^2 + y^2 = 25\)**
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Given that the center of the circle is \((0,0)\) and the radius is \(5\), we can substitute \(h = 0\), \(k = 0\), and \(r = 5\) into the formula:
\[
(x - 0)^2 + (y - 0)^2 = 5^2
\]
This simplifies to:
\[
x^2 + y^2 = 25
\]
Therefore, the correct equation of the circle in standard form is:
**\(x^2 + y^2 = 25\)**