To find the radius of the circle given by the equation:
[tex]\[ 6(x-3)^2 + 6(y+7)^2 = 216 \][/tex]
we need to rewrite this equation in the standard form of a circle equation, which is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Here are the steps to follow:
1. Simplify the given equation:
[tex]\[ 6(x-3)^2 + 6(y+7)^2 = 216 \][/tex]
We can simplify this equation by dividing all terms by 6:
[tex]\[ \frac{6(x-3)^2 + 6(y+7)^2}{6} = \frac{216}{6} \][/tex]
Simplifying, we get:
[tex]\[ (x-3)^2 + (y+7)^2 = 36 \][/tex]
2. Identify the characteristics of the circle:
Now, the equation [tex]\((x-3)^2 + (y+7)^2 = 36\)[/tex] is in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
Here:
- [tex]\( h = 3 \)[/tex]
- [tex]\( k = -7 \)[/tex]
- [tex]\( r^2 = 36 \)[/tex]
These values indicate that the center of the circle is at [tex]\((3, -7)\)[/tex] and that [tex]\( r^2 \)[/tex] (the radius squared) is 36.
3. Determine the radius [tex]\(r\)[/tex]:
To find the radius [tex]\(r\)[/tex], we take the square root of [tex]\(r^2\)[/tex]:
[tex]\[ r = \sqrt{36} = 6 \][/tex]
Therefore, the radius [tex]\(r\)[/tex] of the circle is:
[tex]\[ r = 6 \][/tex]
