If the line [(2-x)/(3)=(3 y-2)/(4 +1)=4-z] makes a right angle with the line [(x+3)/(3 )=(…

Question

If the line $$\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z$$ makes a right angle with the line $$\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}$$, then $$4 \lambda+9 \mu$$ is equal to :

Accepted Answer

$$\begin{aligned}
& L_1: \frac{x-2}{(-3)}=\frac{y-\frac{2}{3}}{\left(\frac{4 \lambda+1}{3}\right)}=\frac{}{(-1)} \
& L_2: \frac{x+3}{3 \mu}=\frac{y-\frac{1}{2}}{-3}=\frac{z-5}{-7} \
& \because L_1 \perp L_2 \
& \Rightarrow(-3)(3 \mu)+\left(\frac{4 \lambda+1}{3}\right)(-3)+(-1)(-7)=0 \
& -9 \mu-4 \lambda-1+7=0 \
& \Rightarrow 4 \lambda+9 \mu=6 \
&
\end{aligned}$$

Solutions · 1