Question

Find the values of a so that the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-a}{4};\frac{x-4}{5}=\frac{y-1}{2}=z$ are skew.

Answer 1

Let $L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-a}{4}$ and; $L_2:\frac{x-4}{5}=\frac{y-1}{2}=z$

The d.r.'s of lines $L_1$ and $L_2$ are respectively 2, 3, 4 and 5, 2, 1

As $2:3:4\ne 5:2:1,\therefore$ the lines are not parallel.

Now coordinates of any random point on the lines $L_1$ and $L_2$ are respectively

$(2\lambda +1,3\lambda +2,\lambda +a)$ and $(5\mu +4,2\mu +1,\mu ).$

Lines will be skew if, apart from being non-parallel, they do not intersect each other. So, there must not exist a pair of values of $\lambda$ and $\mu$ which satisfy the following three equations

simultaneously : $2\lambda +1=5\mu +4,3\lambda +2=2\mu +1,4\lambda +a=\mu$

Solving the first two equations, we get : $\lambda =-1=\mu .$

If these values of $\lambda$ and $\mu$ satisfy the third equation then, the lines will be intersecting each other and hence won't be skew lines.

So, for lines $L_1\&L_2$ to be skew, we must have $4\lambda +a\ne \mu$ i.e., $4\times (-1)+a\ne -1\therefore a\ne 3.$