Answer
(a) The sphere has symmetry with respect to the xy plane.
(b) The sphere has symmetry with respect to this line.
Work Step by Step
We can write the general equation of a sphere:
$(x-a)^2+(y-b)^2+(z-c)^2 = r^2$
where $(a,b,c)$ is the center of the sphere and $r$ is the radius
The equation of the sphere is: $x^2+y^2+z^2 = 25$
The center of the sphere is $(0,0,0)$
(a) A sphere has symmetry with respect to any plane that includes the sphere's center. Since the xy plane includes the point $(0,0,0)$, the sphere has symmetry with respect to the xy plane.
(b) When $n = 1$, the line $(-1,2,3)+n(1,-2,-3)$ passes through the point $(0,0,0)$
A sphere has symmetry with respect to any line that passes through the sphere's center. Since this line passes through the point $(0,0,0)$ which is the sphere's center, the sphere has symmetry with respect to this line.
