To determine the center and radius of the circle given by the equation [tex]\((x-10)^2 + (y-3)^2 = 36\)[/tex], we need to match it to the standard form of a circle's equation:
[tex]\[
(x-h)^2 + (y-k)^2 = r^2
\][/tex]
Here, [tex]\( (h, k) \)[/tex] represents the coordinates of the center of the circle, and [tex]\( r \)[/tex] represents the radius of the circle.
From the given equation:
[tex]\[
(x-10)^2 + (y-3)^2 = 36
\][/tex]
we can see that:
- [tex]\( h = 10 \)[/tex] (the x-coordinate of the center)
- [tex]\( k = 3 \)[/tex] (the y-coordinate of the center)
So, the coordinates of the center of the circle are [tex]\((10, 3)\)[/tex].
Next, we need to find the radius. The given equation is:
[tex]\[
(x-10)^2 + (y-3)^2 = 36
\][/tex]
Here, [tex]\( 36 \)[/tex] represents [tex]\( r^2 \)[/tex] (the square of the radius). To find the radius [tex]\( r \)[/tex], we take the square root of [tex]\( 36 \)[/tex]:
[tex]\[
r = \sqrt{36} = 6
\][/tex]
Therefore, the radius of the circle is [tex]\( 6 \)[/tex].
So, the correct answer is:
[tex]\[
\text{center } =(10, 3), \text{ radius }= 6
\][/tex]
