Question
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 in standard form, using the given information and the pythagorean theorem
Answers
GPT-4o mini
To derive the equation of a circle in standard form, we can use the general formula for the equation of a circle with center \((h, k)\) and radius \(r\):
\[
(x - h)^2 + (y - k)^2 = r^2
\]
From the given information:
- The center of the circle \((h, k)\) is \((2, -2)\).
- The radius \(r\) is \(4\).
Now, substituting the values into the formula:
1. Replace \(h\) with \(2\).
2. Replace \(k\) with \(-2\).
3. Replace \(r\) with \(4\), so \(r^2 = 4^2 = 16\).
The equation becomes:
\[
(x - 2)^2 + (y + 2)^2 = 16
\]
This is the standard form of the equation of the circle.
Thus, the final equation of the circle is:
\[
(x - 2)^2 + (y + 2)^2 = 16
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
From the given information:
- The center of the circle \((h, k)\) is \((2, -2)\).
- The radius \(r\) is \(4\).
Now, substituting the values into the formula:
1. Replace \(h\) with \(2\).
2. Replace \(k\) with \(-2\).
3. Replace \(r\) with \(4\), so \(r^2 = 4^2 = 16\).
The equation becomes:
\[
(x - 2)^2 + (y + 2)^2 = 16
\]
This is the standard form of the equation of the circle.
Thus, the final equation of the circle is:
\[
(x - 2)^2 + (y + 2)^2 = 16
\]