A radio mast stands at P(9, -3, 9), with coordinates in metres…

A radio mast stands at $P(9, -3, 9)$, with coordinates in metres. A straight cable passes through $A(1, -1, 2)$ with direction ratios $(1, 2, 2)$. A technician must attach a sensor to the cable at points that are exactly $15$ metres from the mast. She first finds the shortest distance from the mast to the cable and the foot of the perpendicular. She then uses Pythagoras along the cable to locate the two points. Find the distance of the mast from the cable, the $y$-coordinate of the foot, and the $x$-coordinates of the two points, giving the larger $x$-coordinate first. A third technician wonders whether the answer would change if the cable were described by the direction ratios $(2, 4, 4)$ instead. It would not, because both give the same line.