Let P be the foot of the perpendicular drawn from A(2,3,4) to the given line I. Then,
x−4
−2
=
y
6
=
z−1
−3
Now, any point on the line I is given by x=4−2λ,y=6λ,z=1−3λ The coordinates of P are (4−2λ,6λ,1−3λ) The direction ratios of AP are (4−2λ−2,6λ−3,1−3λ−4) i.e. (2−2λ,6λ−3,−3−3λ) And the direction ratios of I are −2,6 and −3. Given, AP⊥I ∴−2(2−2λ)+6(6λ−3)−3(−3−3λ)=0 ⇒λ=
13
49
∴AP2=(4−2λ−2)2+(6λ−3)2+(1−3λ−4)2 =22−26λ+49λ2 Put λ=