If a line  \(\frac{x+1}{p} = \frac{y-1}{q} = \frac{z-2}{r}\)  where $p=2q=3r$, makes an angle θ with the positive direction of y-axis, then what is cos2θ equal to?

Calculation:

Given,

The line equation is: \( \frac{x+1}{p} = \frac{y-1}{q} = \frac{z-2}{r} \),  where p = 2q = 3r .

The line makes an angle θ  with the positive direction of the y-axis.

Using the relation between the direction cosines

\( l = \frac{p}{\sqrt{p^2 + q^2 + r^2}}, m = \frac{q}{\sqrt{p^2 + q^2 + r^2}}, n = \frac{r}{\sqrt{p^2 + q^2 + r^2}} \),

we substitute p = 3r  and  q = \(\frac{3r}{2} \)

Direction cosine with y -axis is given by \( m = \frac{q}{\sqrt{p^2 + q^2 + r^2}} \).

\( m = \frac{\frac{3r}{2}}{\sqrt{(3r)^2 + \left(\frac{3r}{2}\right)^2 + r^2}} = \frac{\frac{3r}{2}}{\sqrt{9r^2 + \frac{9r^2}{4} + r^2}} = \frac{\frac{3r}{2}}{\sqrt{\frac{49r^2}{4}}} = \frac{3}{7} \).

Now, using the double angle identity for cosine \( \cos 2θ = 2 \cos^2 θ - 1 \):

\( \cos 2θ = 2 \left( \frac{3}{7} \right)^2 - 1 = 2 \times \frac{9}{49} - 1 = \frac{18}{49} - 1 = \frac{-31}{49} \).

∴ The value of \( \cos 2θ \) is \( -\frac{31}{49} \).

Hence, the correct answer is Option 1.