Consider the line (x-2)/2=(2y-5)/(-3), z = -1 . Then which of the fol…

The correct answer is Option (3) → (C) and (D) only

Direction ratios are read from the symmetric form, in which the coefficient of each variable is 1. Dividing the numerator and denominator of the second fraction by 2:

    \( \displaystyle \frac{2y - 5}{-3} = \frac{y - \frac{5}{2}}{-\frac{3}{2}} \)

Since z = −1 is constant along the line, the z-component of the direction is 0. So the line is

    \( \displaystyle \frac{x - 2}{2} = \frac{y - \frac{5}{2}}{-\frac{3}{2}} = \frac{z + 1}{0} \)

with direction ratios \(\left(2, -\frac{3}{2}, 0\right)\). Their magnitude is

    \( \displaystyle \sqrt{4 + \frac{9}{4} + 0} = \sqrt{\frac{25}{4}} = \frac{5}{2} \)

so the direction cosines are

    \( \displaystyle \left(\frac{2}{5/2}, \frac{-3/2}{5/2}, 0\right) = \left(\frac{4}{5}, -\frac{3}{5}, 0\right) \)

So the direction ratios are \(\left(2, -\frac{3}{2}, 0\right)\) and the direction cosines are \(\left(\frac{4}{5}, -\frac{3}{5}, 0\right)\).

Checking the statements

(A) It has Direction ratios (2, −3, −1) — Incorrect. The direction ratios are \(\left(2, -\frac{3}{2}, 0\right)\); the third is 0, since z does not change along the line.

(B) It has Direction cosines \(\left(\frac{4}{5}, \frac{-3}{5}, \frac{-1}{5}\right)\) — Incorrect. The third direction cosine is 0. Also \(\frac{16}{25} + \frac{9}{25} + \frac{1}{25} = \frac{26}{25} \ne 1\).

(C) It has Direction ratios \(\left(2, \frac{-3}{2}, 0\right)\) — Correct. They are the denominators in the symmetric form.

(D) It has Direction cosines \(\left(\frac{4}{5}, \frac{-3}{5}, 0\right)\) — Correct. The direction ratios divided by their magnitude \(\frac{5}{2}\).