Answer:
a) It is the set of point in the the circumference with equation [tex](x-4)^2+y^2=8[/tex].
b) (10.6, 0, -6 )
Step-by-step explanation:
a) The centre of the sphere is (4,0,-6) and the radio of the sphere is [tex]\sqrt{44} \sim 6.6[/tex]. Since |-6|=6 < 6.6, then the sphere intersect the xy-plane and the intersection is a circumference.
Let's find the equation of the circumference.
The equation of the xy-plane is z=0. Replacing this in the equation of the sphere we have:
[tex](x-4)^2+y^2+6^2=44[/tex], then [tex](x-4)^2+y^2=8[/tex].
b) Observe that the point (7,0,-6) has the same y and z coordinates as the centre and the x coordinate of the point is greater than that of the x coordinate of the centre. Then the point of the sphere nearest to the given point will be at a distance of one radius from the centre, in the positive x direction.
(4+[tex]\sqrt{44}[/tex], 0, -6)= (10.6, 0, -6 )
