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A circle is drawn on a coordinate plane with the center point at (2, -2) and a radius of 4. derive the equation of the circle i...Question
A circle is drawn on a coordinate plane with the center point at (0, 0) and a radius of 5. derive the equation of the circle in standard form, using the given information and the pythagorean theorem
Answers
GPT-4o mini
The standard form of the equation of a circle with its center at the point \((h, k)\) and a radius \(r\) is given by the formula:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In your case, the center of the circle is at the origin \((0, 0)\) and the radius \(r\) is 5.
Substituting the values of \(h\), \(k\), and \(r\) into the equation, we have:
- \(h = 0\)
- \(k = 0\)
- \(r = 5\)
Thus, the equation becomes:
\[
(x - 0)^2 + (y - 0)^2 = 5^2
\]
This simplifies to:
\[
x^2 + y^2 = 25
\]
Therefore, the standard form of the equation of the circle is:
\[
x^2 + y^2 = 25
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In your case, the center of the circle is at the origin \((0, 0)\) and the radius \(r\) is 5.
Substituting the values of \(h\), \(k\), and \(r\) into the equation, we have:
- \(h = 0\)
- \(k = 0\)
- \(r = 5\)
Thus, the equation becomes:
\[
(x - 0)^2 + (y - 0)^2 = 5^2
\]
This simplifies to:
\[
x^2 + y^2 = 25
\]
Therefore, the standard form of the equation of the circle is:
\[
x^2 + y^2 = 25
\]