Which of the following cannot be the directi on ratios of the straight line x−3/2 =2−y/3 =z+4/−1?

The correct option is 2,3,−1.

To understand why this is correct, let us analyze the equation of the given straight line and convert it into the standard symmetric form.

The given equation of the line is:

$$\frac{x-3}{2}=\frac{2-y}{3}=\frac{z+4}{-1}$$

Step 1: Convert to Standard Form
The standard symmetric form of a straight line in three-dimensional space is given by:

$$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$$

where $(x_1,y_1,z_1)$ is a point on the line, and $(a,b,c)$ represent the direction ratios of the line. Notice that the coefficients of $x$, $y$, and $z$ in the numerators must all be equal to $+1$.

In the given equation, the term involving $y$ is written as $\frac{2-y}{3}$. To make the coefficient of $y$ equal to $+1$, we multiply both the numerator and the denominator by $-1$:

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

Now, rewriting the entire equation in standard form gives:

$$\frac{x-3}{2}=\frac{y-2}{-3}=\frac{z+4}{-1}$$

Step 2: Identify the Direction Ratios
Comparing this with the standard equation, we see that the base direction ratios of the line are proportional to:
$(a,b,c)=(2,-3,-1)$

Any set of numbers that are proportional to $(2,-3,-1)$ can also serve as the direction ratios of the line. That is, the direction ratios can be represented as $(2k,-3k,-1k)$ for any non-zero real number $k$.

Step 3: Analyze the Options
Let us check each option by finding if a constant multiplier $k$ exists:

1. 2, -3, -1: This matches the base direction ratios directly with $k=1$. Therefore, it can be the direction ratios.

2. -2, 3, 1: Multiplying the base direction ratios by $k=-1$ gives $(-2,3,1)$. Therefore, it can be the direction ratios.

3. 2, 3, -1: Comparing this with the base ratios $(2,-3,-1)$, we see that the y-component has changed sign while the x and z components remained the same. Since there is no constant value $k$ that satisfies:
$2k=2\Rightarrow k=1$
$-3k=3\Rightarrow k=-1$
the set of ratios is not proportional. Therefore, 2, 3, -1 cannot be the direction ratios of the line.

4. 6, -9, -3: Multiplying the base direction ratios by $k=3$ gives $(6,-9,-3)$. Therefore, it can be the direction ratios.