To find the equation of a circle given its center and a point on the circle, we follow these steps:
1. Identify the Center and a Point on the Circle:
- The center of the circle is given as [tex]\((0,0)\)[/tex].
- A point on the circle is given as [tex]\((0,6)\)[/tex].
2. Determine the Radius of the Circle:
- The radius [tex]\( r \)[/tex] of the circle is the distance from the center to any point on the circle.
- Here, we use the given point [tex]\((0,6)\)[/tex] to find the radius.
- The distance between the center [tex]\((0,0)\)[/tex] and the point [tex]\((0,6)\)[/tex] is simply the difference in the y-coordinates since the x-coordinate does not change.
- Therefore, the radius [tex]\( r \)[/tex] is:
[tex]\[
r = 6
\][/tex]
3. Write the General Equation of the Circle:
- The general form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
- For our circle, [tex]\( h = 0 \)[/tex], [tex]\( k = 0 \)[/tex], and [tex]\( r = 6 \)[/tex]. Substituting these values into the general form gives us:
[tex]\[
(x - 0)^2 + (y - 0)^2 = 6^2
\][/tex]
- Simplifying this, we get:
[tex]\[
x^2 + y^2 = 36
\][/tex]
Therefore, the equation of the circle with center [tex]\((0,0)\)[/tex] and passing through the point [tex]\((0,6)\)[/tex] is:
[tex]\[
x^2 + y^2 = 36
\][/tex]
