Answer :
To find the equation of the circle with a given center and circumference, follow these steps:
1. Identify the Given Values:
- The center of the circle is [tex]\((3, 7)\)[/tex]
- The circumference of the circle is [tex]\(8\pi\)[/tex] units
2. Calculate the Radius:
The formula for the circumference [tex]\(C\)[/tex] of a circle is given by:
[tex]\[ C = 2\pi r \][/tex]
where [tex]\(r\)[/tex] is the radius.
We can rearrange this formula to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \frac{C}{2\pi} \][/tex]
Substitute the given circumference [tex]\(8\pi\)[/tex] into the formula:
[tex]\[ r = \frac{8\pi}{2\pi} = 4 \][/tex]
Thus, the radius [tex]\(r\)[/tex] is 4 units.
3. Formulate the Standard Equation of the Circle:
The standard form for 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]\((3, 7)\)[/tex] and the calculated radius [tex]\(4\)[/tex] into the equation:
[tex]\[ (x - 3)^2 + (y - 7)^2 = 4^2 \][/tex]
Simplify the right side of the equation by squaring the radius:
[tex]\[ 4^2 = 16 \][/tex]
4. Write the Final Equation:
The equation of the circle is:
[tex]\[ (x - 3)^2 + (y - 7)^2 = 16 \][/tex]
This is the equation of the circle with center [tex]\((3, 7)\)[/tex] and circumference [tex]\(8\pi\)[/tex] units.
1. Identify the Given Values:
- The center of the circle is [tex]\((3, 7)\)[/tex]
- The circumference of the circle is [tex]\(8\pi\)[/tex] units
2. Calculate the Radius:
The formula for the circumference [tex]\(C\)[/tex] of a circle is given by:
[tex]\[ C = 2\pi r \][/tex]
where [tex]\(r\)[/tex] is the radius.
We can rearrange this formula to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \frac{C}{2\pi} \][/tex]
Substitute the given circumference [tex]\(8\pi\)[/tex] into the formula:
[tex]\[ r = \frac{8\pi}{2\pi} = 4 \][/tex]
Thus, the radius [tex]\(r\)[/tex] is 4 units.
3. Formulate the Standard Equation of the Circle:
The standard form for 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]\((3, 7)\)[/tex] and the calculated radius [tex]\(4\)[/tex] into the equation:
[tex]\[ (x - 3)^2 + (y - 7)^2 = 4^2 \][/tex]
Simplify the right side of the equation by squaring the radius:
[tex]\[ 4^2 = 16 \][/tex]
4. Write the Final Equation:
The equation of the circle is:
[tex]\[ (x - 3)^2 + (y - 7)^2 = 16 \][/tex]
This is the equation of the circle with center [tex]\((3, 7)\)[/tex] and circumference [tex]\(8\pi\)[/tex] units.
