Question
A circle is drawn on a coordinate plane with the center point at (5,−3) and a radius of 6. 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 use the general formula for a circle given its center and radius. The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the center of the circle is given as \((5, -3)\) and the radius is \(6\).
Here, we can identify:
- \(h = 5\)
- \(k = -3\)
- \(r = 6\)
Now, we square the radius:
\[
r^2 = 6^2 = 36
\]
Substituting these values into the standard form equation, we have:
\[
(x - 5)^2 + (y + 3)^2 = 36
\]
Thus, the equation of the circle in standard form is:
\[
(x - 5)^2 + (y + 3)^2 = 36
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the center of the circle is given as \((5, -3)\) and the radius is \(6\).
Here, we can identify:
- \(h = 5\)
- \(k = -3\)
- \(r = 6\)
Now, we square the radius:
\[
r^2 = 6^2 = 36
\]
Substituting these values into the standard form equation, we have:
\[
(x - 5)^2 + (y + 3)^2 = 36
\]
Thus, the equation of the circle in standard form is:
\[
(x - 5)^2 + (y + 3)^2 = 36
\]