What is the area of the region defined by the equation \(x^2+y^2 - 7 = 2y-8x+1?\)
Re-arrange:
x^2 +8x + y^2 - 2y = 8 complete the squares for x and y
(x+4)^2 + (y-1)^2 = 8 + 16 +1
(x+4)2 + (y-1)2 = 52 this is circle equation with radius = 5
pi r2 = area of circle
pi 52 = 25 pi units2 ( 78.54 units2)
x^2 + y^2 = - 7 = 2y - 8x + 1
x^2 + 8x + y^2 - 2y = 7 + 1 complete the square on x and y
x^2 + 8x + 16 + y^2 - 2y + 1 = 7 + 1 + 16 + 1
(x + 4)^2 + ( y - 1)^2 = 25
This is a circle centered at ( -4, 1) with a radius of 5
The area = pi 5^2 = 25 pi
