Let D = (0,0) Let B = (2,8)
We can find the coordinates of F thusly :
Construct a circle with a radius of 2 centered at D
The equation of this circle is x^2 + y^2 = 4 (1)
Construct a circle with a radius of 8 centered at B
The eqation of this circle is ( x - 2)^2 + (y - 8)^2 = 64 ⇒ x^2 - 4x + 4 + y^2 - 16y + 64 = 64 ⇒
x^2 - 4x +y^2 - 16y = -4 (2)
Subtract (2) from (1)
4x + 16y = 8
x + 4y = 2
x = 2 - 4y sub this into ( 1) for x
(2 - 4y)^2 + y^2 = 4
16y^2 - 16y + 4 + y^2 = 4
17y^2 - 16y = 0
y ( 17y - 16) = 0
Solving for the second gives the y coordinate of F y = 16/17
And x = 2 - 4(16.17) = [34 - 64]/ 17 = -30/17
F = (-30/17 , 16/17)
And the slope of BF = [ 8 - 16/17 ] / [ 2 - -30/17 ] = (120/17) / (64/17) = 120/64 = 15/8
So the equation of a line through BF is
y = (15/8)(x - 2) + 8
y = (15/8)x - 30/8 + 8
y = (15/8)x - 15/4 + 8
y = (15/8)x + 17/4
So when x = 0 this line intersects the left side of the rectangle at 17/4 = 4.25
So we have a right triangle with a leg of (8 - 4.25) and 2 = 3.75 and 2
And the area of two of these triangles is 3.75 * 2 = 7.5 (3)
So...the gray area = area of rectangle - (3) = 8*2 - 7.5 = 16 - 7.5 = 8.5