Question
Find the equation of the perpendicular drawn from the point (2, 4, – 1) to the line 
Solution
Let the given line AB be
Any point M on line is (r – 5, 4 r – 3, – 9 r + 6)
Let this point M be the foot of perpendicular from P(2, 4, –1) on AB.
Any point M on line is (r – 5, 4 r – 3, – 9 r + 6)
Let this point M be the foot of perpendicular from P(2, 4, –1) on AB.
Direction ratios of PM are
r – 5 – 2, 4 r – 3 – 4, – 9 r + 6 + 1 i.e. r – 7, 4 r – 7, – 9 r + 7
Direction ratios of AB are 1, 4, – 9
Since PM ⊥ AB
∴ (1) (r – 7) + (4) (4 r – 7) + (– 9) (–9 r + 7) = 0
∴ r – 7 + 16 r – 28 + 81 r – 63 = 0
∴ 98 r = 98 ⇒ r = 1
∴ Mis (1 – 5, 4 – 3, – 9 + 6) i.e. (– 4, 1, –3)
The equation of PM is
i.e., 
i.e., 
