Consider the line $\frac{x-2}{2} = \frac{2y-5}{-3}, z = -1$. Then which of the following is/are true?

(A) It has Direction ratios (2, -3, -1)

(B) It has Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, \frac{-1}{5}\right)$

(C) It has Direction ratios $\left(2, \frac{-3}{2}, 0\right)$

(D) It has Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, 0\right)$

Choose the correct answer from the options given below:

The line is given as $\frac{x-2}{2} = \frac{2y-5}{-3}, z = -1$

The standard form of a line in 3D is $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$ where $(a, b, c)$ are the direction ratios.

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The y-term can be rewritten:

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

$= \frac{y-\frac{5}{2}}{-\frac{3}{2}}$

Since $z = -1$ is constant, the line doesn't move in the z-direction. This is written as:

$\frac{z-(-1)}{0} = \frac{z+1}{0}$

When a coordinate is constant, its direction ratio is 0.

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The line in standard form:

$\frac{x-2}{2} = \frac{y-\frac{5}{2}}{-\frac{3}{2}} = \frac{z+1}{0}$

Direction Ratios $= \left(2, -\frac{3}{2}, 0\right)$

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Direction cosines are normalized direction ratios:

$\text{Direction cosines} = \left(\frac{a}{\sqrt{a^2+b^2+c^2}}, \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2+c^2}}\right)$

For $(a, b, c) = \left(2, -\frac{3}{2}, 0\right)$:

$\sqrt{a^2+b^2+c^2} = \sqrt{(2)^2 + \left(-\frac{3}{2}\right)^2 + (0)^2}$

$= \sqrt{4 + \frac{9}{4} + 0}$

$= \sqrt{\frac{16+9}{4}}$

$= \sqrt{\frac{25}{4}}$

$= \frac{5}{2}$

Direction cosines:

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

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(A) Direction ratios $(2, -3, -1)$ - Incorrect. The z-direction ratio is 0, not -1, and y-direction ratio is $-\frac{3}{2}$, not -3.

(B) Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, \frac{-1}{5}\right)$ - Incorrect. The z-direction cosine is 0, not $-\frac{1}{5}$.

(C) Direction ratios $\left(2, \frac{-3}{2}, 0\right)$ - Correct.

(D) Direction cosines $\left(\frac{4}{5}, \frac{-3}{5}, 0\right)$ - Correct.

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Therefore, options (C) and (D) are true.