Any point P on the line
\(\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9} = \lambda\)
is \(P(\lambda - 5, 4\lambda - 3, 6-9\lambda)\)
\(\because\) it is given that the distance between point P and Q is 7 units.
\(\Rightarrow |PQ| = 7\)
\(= |PQ|^2 = 49\)
Now \(\vec{PQ} = (\lambda - 7)\hat{i} + (4\lambda - 7)\hat{j} +(7-9\lambda)\hat{k}\)
\(\Rightarrow (\lambda - 7)^2 + (4\lambda - 7)^2+(7-9\lambda)^2 = 49\)
\(\Rightarrow \lambda = 1\)
\(\therefore\) point P will be (–4, 1, –3)
The equation of line joining (–4, 1, –3) and (2, 4, –1) will have direction ratio \(<6,3,2>\)
\(\therefore\) Line joining P & Q is
\(\frac{x-2}{6} = \frac{y-4}{3} = \frac{z+1}{2}\)
