Let the line segment formed by joining the points (1,−3,−6) and (−2,4,−7) is divided by the XY plane in the ratio k:1 at the point P, then the co-ordinates of P are
(−2k+1k+1,4k−3k+1,−7k−6k+1)
But P lies in the XY plane,
∴z−co-ordinate is zero
⇒−7k−6k+1=0
⇒−7k−6=0
⇒7k=−6
⇒k=−67
Hence,XY plane divides the line segment AB in the ratio −67:1
i.e.,6:7 externally and the coordinates of the point of division are
⎛⎜
⎜
⎜⎝−2×−67+1−67+1,4×−67−3−67+1,−7×−67−6−67+1⎞⎟
⎟
⎟⎠
or ⎛⎜
⎜
⎜⎝12+77−6+77,−24−217−6+77,42−427−67+1⎞⎟
⎟
⎟⎠
or (19,−45,0)