If the lines (1-x)/2=displaystyle2y/p=(z-1)/1 and (3-2x)/p=(y-1)/2=di…

The correct answer is Option (2) → 4

Direction ratios can be read only from the standard form \( \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c} \), where the coefficients of x, y, z in the numerators are 1.

First line:

    \( \displaystyle \frac{1-x}{2}=\frac{x-1}{-2} \)

    \( \displaystyle \frac{2y}{p}=\frac{y}{p/2} \)

    \( \displaystyle \frac{x-1}{-2}=\frac{y-0}{p/2}=\frac{z-1}{1} \)

So the direction ratios are

    \( \displaystyle \left(-2,\ \frac{p}{2},\ 1\right) \)

Second line:

    \( \displaystyle \frac{3-2x}{p}=\frac{-2\left(x-\frac{3}{2}\right)}{p} \)

    \( \displaystyle =\frac{x-\frac{3}{2}}{-p/2} \)

    \( \displaystyle \frac{x-\frac{3}{2}}{-p/2}=\frac{y-1}{2}=\frac{z-0}{1} \)

So the direction ratios are

    \( \displaystyle \left(-\frac{p}{2},\ 2,\ 1\right) \)

Parallel lines have proportional direction ratios. The z-ratios are both 1, so the ratios must be equal:

    \( \displaystyle -2=-\frac{p}{2}\ \Rightarrow\ p=4 \)

    \( \displaystyle \frac{p}{2}=2\ \Rightarrow\ p=4 \)

Both conditions give p = 4.

Tip: Always make the coefficients of x, y and z equal to +1 before reading direction ratios: \( \frac{1-x}{2} \) has direction ratio −2, and \( \frac{3-2x}{p} \) has −p/2.