Find the equation x2 + y2 + Dx + Ey + F = 0

Question

Find the equation x² + y² + Dx + Ey + F = 0 of the circle that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle.



(0, 0), (0, −4), (5, 0)

Answer 1

circle's equation with circle through points (0,0), (0,4), and (-5,0)

x^2 +y^2 + dx +ey +f = 0

plug in the origin to get f = 0

then plug in (0,4) to get

16 + 4e = 0

e = -4

then plug in (-5,0)

25 -5d = 0

d = 25/5 =5

x^2 +y^2 +5x -4y = 0

or

x^2 +5x + 25/4 + y^2 -4y +4 = 25/4 +4

(x+2.5)^2 + (y-2)^2 = 41/4

is a circle with center (-2.5, 2) and radius squared = 41/4, radius = (sqr41)/2

Answer 2

Plug in the x,y values of the points and get 3 simultaneous equations for D, E, and F

(0,0): F = 0

(0,-4): 16 -4E = 0 E = 4

(5,0): 25 + 5D = 0 D = -5

Points are selected to make things easier.