ANALYTIC GEOMETRY

The circle has a radius of 12, and the line y = 6 - x splits it in half, so the area is 1/2*12^2*pi = 72*pi.

See the graph  here :    https://www.desmos.com/calculator/qeidqnvtts

The intersection of this line and the circle  will form the diameter of the  circle....to find the radius we can  complete the square  on x  in the equation of the circle......so we have....

x^2 - 12x  +36  + y^2   = 28 + 36

(x + 6)^2  + y^2 =  64

The center is  (6, 0)

The radius  = sqrt (64)   = 8

So....the area we are looking for  is

(1/2) pi (8)^2 =   

32 pi units^2

Thanks Chris, I think you overlooked one of the conditions.

I just did this in ful but it was lost.

Anyway here is my pic

\(Area = \frac{135}{360}\times \pi\times 6^2\\ Area = \frac{27\pi }{2} \; units\; squared \)

GeoNewbie21  has just  pointed out to me that the radius is 8 not 6

So the answer should have been ....

\(Area = \frac{135}{360}\times \pi\times \color{red}{8}^2\\ Area = \color{red}24\pi \color{black}\; units\; squared\)

Thanks Geo