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
Answers
GPT-4o mini
To write the equation of a circle in standard form, you can use the formula:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle, and \(r\) is the radius.
From the given information:
- The center of the circle is at the point \((-1, 3)\), so \(h = -1\) and \(k = 3\).
- The radius of the circle is \(r = 5\).
Now, we can substitute these values into the standard form equation:
\[
(x - (-1))^2 + (y - 3)^2 = 5^2
\]
This simplifies to:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Thus, the equation of the circle in standard form is:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle, and \(r\) is the radius.
From the given information:
- The center of the circle is at the point \((-1, 3)\), so \(h = -1\) and \(k = 3\).
- The radius of the circle is \(r = 5\).
Now, we can substitute these values into the standard form equation:
\[
(x - (-1))^2 + (y - 3)^2 = 5^2
\]
This simplifies to:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]
Thus, the equation of the circle in standard form is:
\[
(x + 1)^2 + (y - 3)^2 = 25
\]