Basic Manipulation.

Applying Cauchy-Schwarz Inequality $\displaystyle ({3}^2 + {4}^2 + {5}^2)( {x}^2 + {y}^2 + {z}^2) \geq {(3x + 4y + 5z)}^2 = {(10\sqrt{2})}^2$ Hence $\displaystyle 50({x}^2 + {y}^2 + {z}^2)\geq 200$ . Hence the minimum value is 4. Answer

If $\pi$ is a given plane and $P$ is a given point then the the point on plane which is nearest to $P$ is the foot of the perpendicular from $P$ to the plane. $:)$

here is an innovative way

3x+4y+5z=10(2^0.5)

(3i+4j+5k) . (xi+yj+zk) = 10(2^0.5)...dot product

(xx+yy+zz)(50) cos(t) cos(t)= 200...take dot product the other way and square

for min cost =1

and answer is 4