4x²+4y²+4z²-12x+8y-16z+13=0,
4x²-12x+9+4y²+8y+4+4z²-16z+16-29+13=0,
4(x-3/2)²+4(y+1)²+4(z-2)²-16=0,
(x-3/2)²+(y+1)²+(z-2)²=4² is a sphere of radius 4, centre (3/2,-1,2).
The plane containing the given points can be represented:
Ax+By+Cz+D=0. Plugging in the points:
B+C+D=0, A+C+D=0, A+B+D=0, where A, B, C, D are constants.
We can write this system as:
B+C=A+C=A+B=-D. Therefore A=B=C=-D/2 and the equation of the plane is:
(-x-y-z)/2+1=0, x+y+z=2.
All parallel planes will have the form: x+y+z=p where p is a constant.
To pass through the centre of the sphere:
3/2-1+2=5/2=p, making the equation of the required plane:
x+y+z=5/2.
