Question
A circle is drawn on a coordinate plane with the center point at (-2, 0) and a radius of 7. 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 following formula:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle, and \(r\) is the radius.
Given the center of the circle at \((-2, 0)\) and a radius of \(7\):
1. Identify \(h\), \(k\), and \(r\):
- \(h = -2\)
- \(k = 0\)
- \(r = 7\)
2. Substitute these values into the standard form equation:
\[
(x - (-2))^2 + (y - 0)^2 = 7^2
\]
This simplifies to:
\[
(x + 2)^2 + y^2 = 49
\]
Thus, the equation of the circle in standard form is:
\[
(x + 2)^2 + y^2 = 49
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle, and \(r\) is the radius.
Given the center of the circle at \((-2, 0)\) and a radius of \(7\):
1. Identify \(h\), \(k\), and \(r\):
- \(h = -2\)
- \(k = 0\)
- \(r = 7\)
2. Substitute these values into the standard form equation:
\[
(x - (-2))^2 + (y - 0)^2 = 7^2
\]
This simplifies to:
\[
(x + 2)^2 + y^2 = 49
\]
Thus, the equation of the circle in standard form is:
\[
(x + 2)^2 + y^2 = 49
\]