Question

If the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then which of the following statement(s) is/are correct ?
(where $[.]$ denotes $G.I.F.$)

• A. $4k-17=0$
• B. $3k-14=0$
• C. $\left[k\right]=4$
• D. $4k^2-10k-9\left[k\right]=0$

Answer 1

Answer: (C), (D)

$4k^2-10k-9\left[k\right]=0$

Given : line $L_1:\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ passese through point $A(1,-1,1)$ and parallel to vector $2\hat{i}+3\hat{j}+4\hat{k}$
line $L_2:\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ passes through point $B(3,k,0)$ and parallel to vector $\hat{i}+2\hat{j}+\hat{k}$
As the given lines are not parallel, the are intersecting if vector $\overrightarrow{AB},2\hat{i}+3\hat{j}+4\hat{k},\hat{i}+2\hat{j}+\hat{k}$ are coplanar
$\therefore \left|\begin{array}{ccc}3-1& k+1& 0-1\\ 2& 3& 4\\ 1& 2& 1\end{array}\right|=0$
$\Rightarrow 2(k+1)=11\Rightarrow k=\frac{9}{2}$