Given lines

$${L_1}:r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k),\,\lambda \in R$$ and

$${L_2}:r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R$$

and since line L₃ is perpendicular to both lines L₁ and L₂.

Then a vector along L₃ will be,

$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr { - 1} & 2 & 2 \cr 2 & { - 1} & 2 \cr } } \right| = \widehat i(4 + 2) - \widehat j( - 2 - 4) + \widehat k(1 - 4)$$

$$ = 6\widehat i + 6\widehat j - 3\widehat k = 3(2\widehat i + 2\widehat j - \widehat k)$$ ....(i)

Now, let a general point on line L₁.

$$P(1 - \lambda ,\,2\lambda ,\,2\lambda )$$ and on line L₂ as $$Q(2\mu - \mu ,\,2\mu )$$ and let P and Q are point of intersection of lines L₁, L₃ and L₂, L₃, so direction ratio's of L₃

$$(2\mu + \lambda - 1,\, - \mu - 2\lambda ,\,\,2\mu - 2\lambda )$$ ....(ii)

Now, $${{2\mu + \lambda - 1} \over 2} = {{\, - \mu - 2\lambda } \over 2}\, = {{\,2\mu - 2\lambda } \over 1}$$

[from Eqs. (i) and (ii)]

$$ \Rightarrow \lambda = {1 \over 9}$$ and $$\mu = {2 \over 3}$$

So, $$P\left( {{8 \over 9},{2 \over 9},{2 \over 9}} \right)$$ and $$Q\left( {{4 \over 9}, - {2 \over 9},{4 \over 9}} \right)$$

Now, we can take equation of line L₃ as $$r = a + t(2\widehat i + 2\widehat j - \widehat k)$$, where a is position vector of any point on line L₃ and possible vector of a are

$$\left( {{8 \over 9}\widehat i + {2 \over 9}\widehat j + {2 \over 9}\widehat k} \right)$$ or $$\left( {{4 \over 9}\widehat i - {2 \over 9}\widehat j + {4 \over 9}\widehat k} \right)$$ or $$\left( {{2 \over 3}\widehat i + {1 \over 3}\widehat k} \right)$$

Hence, options (a), (b) and (c) are correct.