The sphere has symmetry with respect to this plane.
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: $(x-1)^2+(y-2)^2+(z+5)^2 = 49$ The center of the sphere is $(1,2,-5)$ We can verify if the plane includes the sphere's center: $3x-4y+5z = 3(1)-4(2)+5(-5) = -30$ Since the sphere's center satisfies the conditions of the plane's equation, the sphere's center is included in the plane. A sphere has symmetry with respect to any plane that includes the sphere's center. Since this plane includes the point $(1,2,-5)$ which is the sphere's center, the sphere has symmetry with respect to this plane.