Question

If the straight lines $\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}$ intersect at a point then the integer $k$ is equal to

• A. $-5$
• B. $5$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $-5$

The lines are :

$\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}$

Since, lines intersect at a point, shortest distance b/w them is 0.

$\left|\begin{array}{ccc}k& 2& 3\\ 3& k& 2\\ 1& 1& -2\end{array}\right|=0$

$k(-2k-2)-2(-6-2)+3(3-k)=0$

$-2k^2-5k+25=0$

$2k^2+5k-25=0$

$2k^2+10k-5k-25=0$

$2k(k+5)-5(k+5)=0$

$(k+5)(2k-5)=0$

$k=-5,\frac{5}{2}$

Hence, integer value is -5.