Question

Find the distance between the spheres x^2 +y^2 + Z^2 -2x-6y+8z+17=0 and 3x^2+3y^2+3z^2+12x-6y-18z+36=0

Answer 1

x^2 +y^2 + Z^2 -2x-6y+8z+17=0

Simplify this equation.

x^2-2x+1+y^2-6y+9+z^2+8z+16 +17=1+9+16

(x-1)^2+(y-3)^2+(z+4)^2=9

(x-1)^2+(y-3)^2+(z+4)^2=3^2

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3x^2+3y^2+3z^2+12x-6y-18z+36=0

Simplify this equation.

x^2+y^2+z^2+4x-2y-6z+12=0

x^2+4x+y^2-2y+z^2-6z+12=0

x^2+4x+4+y^2-2y+1+z^2-6z+9+12=4+1+9

(x+2)^2+(y-1)^2+(z-3)^2=2

(x+2)^2+(y-1)^2+(z-3)^2=(√2)^2

Centers are C_1 = (1,3,-4) and C_2 = (-2,1,3)

Radii are r_1 =3 and r_2 =√2

Calculate the distance between centers.

C_1C_2=√[(-2-1)^2+(1-3)^2+(3-(-4))^2]=√(9+4+49) =√62

Calculate the shortest distance.

d=C_1C_2-r_1-r_2 =√62-3-√2=3.46 units