Answer :
To write the equation of the circle in standard form, we need to know the center and the radius of the circle.
Step 1: Identify the center of the circle
The center of the circle is given as [tex]\((10, 7)\)[/tex].
Step 2: Determine the radius of the circle
We know the circumference of the circle, which is given as [tex]\(14\pi\)[/tex]. The formula for the circumference [tex]\(C\)[/tex] of a circle in terms of the radius [tex]\(r\)[/tex] is:
[tex]\[ C = 2\pi r \][/tex]
Given [tex]\( C = 14\pi \)[/tex], we can solve for the radius [tex]\( r \)[/tex] by dividing both sides of the equation by [tex]\(2\pi\)[/tex]:
[tex]\[ r = \frac{C}{2\pi} = \frac{14\pi}{2\pi} = 7 \][/tex]
So, the radius of the circle is [tex]\(7\)[/tex].
Step 3: Write the equation of the circle in standard form
The standard 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]
Substitute the given center [tex]\((h, k) = (10, 7)\)[/tex] and the radius [tex]\(r = 7\)[/tex]:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 7^2 \][/tex]
Simplify [tex]\(7^2\)[/tex] to get [tex]\(49\)[/tex]:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 49 \][/tex]
Thus, the equation of the circle in standard form is:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 49 \][/tex]
Step 1: Identify the center of the circle
The center of the circle is given as [tex]\((10, 7)\)[/tex].
Step 2: Determine the radius of the circle
We know the circumference of the circle, which is given as [tex]\(14\pi\)[/tex]. The formula for the circumference [tex]\(C\)[/tex] of a circle in terms of the radius [tex]\(r\)[/tex] is:
[tex]\[ C = 2\pi r \][/tex]
Given [tex]\( C = 14\pi \)[/tex], we can solve for the radius [tex]\( r \)[/tex] by dividing both sides of the equation by [tex]\(2\pi\)[/tex]:
[tex]\[ r = \frac{C}{2\pi} = \frac{14\pi}{2\pi} = 7 \][/tex]
So, the radius of the circle is [tex]\(7\)[/tex].
Step 3: Write the equation of the circle in standard form
The standard 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]
Substitute the given center [tex]\((h, k) = (10, 7)\)[/tex] and the radius [tex]\(r = 7\)[/tex]:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 7^2 \][/tex]
Simplify [tex]\(7^2\)[/tex] to get [tex]\(49\)[/tex]:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 49 \][/tex]
Thus, the equation of the circle in standard form is:
[tex]\[ (x - 10)^2 + (y - 7)^2 = 49 \][/tex]
