Question

Q4 Determine the coordinates of the centre of a
sphere that passes through points (0,6, 2),
(4,3, 2), and (1,4, 2) and whose centre
passes through the plane x + y + z = 0.
Ops: A.
(3.5, 6.5, 5)
(2.5, 5.5,-5)
C.
(2.5, 5.5, 10)
D.
(3.5, 6.5, -10)​

Answer 1

Since the work behid it is heavy i'll only leave you hints.

• The general equation of a sphere can be obtained by extrapolating the circle's equation into three dimensions.
• That is,
• $x^{2} + y^{2} + z^{2} + 2gx +2fy + 2hz + c = 0$ and (-g,-f,-h) is the coordinates of the center. You can derive this if you work onto what defines a sphere!
• Now we have three points, and by assuming that they are not coplanar, they must be lying on one sphere.
• Hence, you plug the points onto the gen. equation to get 3 equations which has 4 Unknowns(g,f,h,c)
• Now we have other clue which gives the relation between (g,f,h) that is it lies in the plane x+y+z=0. Therefore, g = - (h + f)
• Yeah we solved it now as you realise, plug the value of g in  those three equations to get 3 Equations with 3 Unknowns and solve it by matrix method or cramers rule or whatever method you like!!