Solution

Above formula is used to find angle $$\theta$$ between two lines having direction ratios $$a_1,b_1,c_1$$ and $$a_2,b_2,c_2$$

Now let us find the direction cosines of the given lines

$$\dfrac{x}{2}=\dfrac{y}{2}=\dfrac{z}{1}$$ direction cosines are $$2,2,1$$

next line can also be written as
$$\dfrac{x-5}{2}=\dfrac{y-2}{\dfrac{P}{7}}=\dfrac{z-3}{4}$$
hence direction cosines are $$2,\dfrac{P}7,4$$

and we know that $$cos\theta=\dfrac23$$

Keeping the value of these cosines in above formula we get

$$\dfrac{2}{3}=\left|\dfrac{2\times 2+2\times \dfrac{P}{7}+1\times 4}{\sqrt{2^2+2^2+1^2}\sqrt{2^2+\dfrac{P^2}{49}+4^2}}\right|=\dfrac{8+\dfrac{2P}{7}+4}{3\times \sqrt{2^2+\dfrac{P^2}{49}+4^2}}$$

after cross multiplication we get

$$\Rightarrow \left(4+\dfrac{P}{7}\right)^2=20+\dfrac{P^2}{49}$$

$$\Rightarrow 16+\dfrac{8P}{7}+\dfrac{P^2}{49}=20+\dfrac{P^2}{49}$$

$$\Rightarrow \dfrac{8P}{7}=4$$

$$\Rightarrow P=\dfrac{7}{2}$$

Therefore Answer is $$D$$