Question

Write the equation of a circle with center (4,5) and a Circumference = 16 pi

Answer 1

Equation of the circle : (x - 4)² + (y - 5)² = 64

Explanation:

Equation of the circle

(x - x₁)² + (y - y₁)² = r²

Where (x₁, y₁) be the coordinates of center and r is the radius of circle.

To find the radius

It is given that circumference of circle = 16π

Circumference 2πr = 16

r = 16π/2π= 8

To find the equation of the circle

Center = (4, 5)

(x - x₁)² + (y - y₁)² = r²

(x - 4)² + (y - 5)² = 8²

(x - 4)² + (y - 5)² = 64

Answer 2

The required equation in standard form is
`(x-4)^2+(y-5)^2=64`

Explanation:

The equation of a circle with center (h,k) an radius, r units is given by the formula;

`(x-h)^2+(y-k)^2=r^2`

The given circle has center (4,5) and the radius can be calculated from the given circumference, which is
`C=16\pi`

`2\pi r=16\pi`

`\implies r=8`

We substitute these values into the formula to obtain;

`(x-4)^2+(y-5)^2=8^2`

We simplify to get;

`(x-4)^2+(y-5)^2=64`