Question
Find the length of the perpendicular drawn from the point (1, ,2 3) on the line
Solution
Let the given line AB be
Any point M on this line is
(3 r + 6, 2 r + 7, –2 r + 7)
Let this point M be the foot of perpendicular from P(1, 2, 3) on AB
Direction ratios of PM are
3 r + 6 – 1, 2 r + 7 – 2, – 2 r + 7 – 3 i.e. 3 r + 5, 2 r + 5, – 2 r + 4
Direction ratios of AB are 3, 2, –2
Since PM ⊥ AB
∴ (3 r + 5) (3) + (2 r + 5) (2) + (– 2 r + 4) (–2) = 0
∴ 9 r + 15 + 4 r + 10 + 4 r – 8 = 0
⇒ 17 r = –17 ⇒ r = – 1
∴ M is (3, 5, 9)
∴ required length of perpendicular = PM
