Equation of a Plane

Question

Find an equation of the plane that contains both the center and  "north pole" of the sphere

   (x-2)^2+(y-1)^2+(z-3)^2 = 64

and is parallel to the line given in vector form by r(t) = [5*t+63, -54*t+47, 54*t-89].  Write your answer in the form ax + by + cz = d

Answer 1

The center is (2,1,3)

the plane contains the vector (0,0,1) (because it contains the northpole) and the vector (5 , -54, 54)  (because it is parallel to the line)

 

The normal vector (a,b,c) is orthogonal to the others vectors, then :

c = 0

and 5a - 54b = 0

then if a = 54, b = 5 and the equation is

54x + 5y = d

the center is into the plane so :

54*2 + 5*1 = d

d = 113

the equation is 54x + 5y = 113