Question
The line which passes through the origin and intersect the two lines $$\frac{{x - 1}}{2} = \frac{{y + 3}}{4} = \frac{{z - 5}}{3},\,\frac{{x - 4}}{2} = \frac{{y + 3}}{3} = \frac{{z - 14}}{4},{\text{ is :}}$$
A.
$$\frac{x}{1} = \frac{y}{{ - 3}} = \frac{z}{5}$$
B.
$$\frac{x}{{ - 1}} = \frac{y}{3} = \frac{z}{5}$$
C.
$$\frac{x}{1} = \frac{y}{3} = \frac{z}{{ - 5}}$$
D.
$$\frac{x}{1} = \frac{y}{4} = \frac{z}{{ - 5}}$$
Answer :
$$\frac{x}{1} = \frac{y}{{ - 3}} = \frac{z}{5}$$
Solution :
Let the line be $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}......\left( {\text{i}} \right)$$
If line $$\left( {\text{i}} \right)$$ intersects with the line $$\frac{{x - 1}}{2} = \frac{{y + 3}}{4} = \frac{{z - 5}}{3},$$ then
\[\left| \begin{array}{l}
a\,\,\,\,\,\,\,\,b\,\,\,\,\,c\\
2\,\,\,\,\,\,\,\,4\,\,\,\,\,\,3\\
4\,\,\, - 3\,\,\,\,\,14\,
\end{array} \right| = 0 \Rightarrow 9a - 7b - 10c = 0......\left( {{\rm{ii}}} \right)\]
From $$\left( {\text{i}} \right)$$ and $$\left( {\text{i}} \right),$$ we have $$\frac{a}{1} = \frac{b}{{ - 3}} = \frac{c}{5}$$
$$\therefore $$ The line is $$\frac{x}{1} = \frac{y}{{ - 3}} = \frac{z}{5}$$