To write the equation of a circle in standard form, we need to identify the center and radius of the circle.
Given:
- The center of the circle is (10, 7).
- The circumference is 14π.
First, we use the given circumference to find the radius of the circle. The circumference [tex]\( C \)[/tex] of a circle is given by the formula:
[tex]\[ C = 2 \pi r \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle. Given that [tex]\( C = 14 \pi \)[/tex], we can solve for [tex]\( r \)[/tex] as follows:
[tex]\[ 14 \pi = 2 \pi r \][/tex]
Divide both sides by [tex]\( 2 \pi \)[/tex]:
[tex]\[ r = \frac{14 \pi}{2 \pi} = 7 \][/tex]
Now that we have the radius of the circle as 7, the equation of a circle in standard form is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\( r \)[/tex] is the radius. Plugging in the center [tex]\((10, 7)\)[/tex] and the radius [tex]\(7\)[/tex], we get:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 7^2 \][/tex]
Calculate [tex]\( 7^2 \)[/tex]:
[tex]\[ 7^2 = 49 \][/tex]
Therefore, the equation of the circle in standard form is:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 49 \][/tex]
