A line with direction ratios 1,-1,2 intersects the lines (x)/(2)=(y)/(3)=(z+1)/(3) and (x+…

Question

A line with direction ratios $1,-1,2$ intersects the lines $\frac{x}{2}=\frac{y}{3}=\frac{z+1}{3}$ and $\frac{x+1}{-1}=\frac{y-2}{1}=\frac{z}{4}$ at the points P and Q , respectively. If the length of the line segment PQ is $\alpha$

Accepted Answer

P$(2\lambda ,3\lambda ,3\lambda -1)$ and Q$(-\mu -1,\mu +2,4\mu )$

D.R of PQ $2\lambda +\mu +1$, $3\lambda -\mu -2$, $3\lambda -4\mu -1$,

$\frac{2\lambda +\mu +1}{1}=\frac{3\lambda -\mu -2}{-1}=\frac{3\lambda -4\mu -1}{2}$

Solving $\lambda =\frac{1}{5}$ and $\mu =-\frac{8}{15}$

P$(\frac{2}{5},\frac{3}{5},-\frac{2}{5})$ \& Q$(-\frac{7}{15},\frac{22}{15},-\frac{32}{15})$

${PQ}^{2}={(\frac{13}{15})}^{2}+{(\frac{13}{15})}^{2}+{(\frac{26}{15})}^{2}$

${\alpha }^{2}=\frac{1014}{225}	herefore 225{\alpha }^{2}=1014$

Solutions · 1