The straight line `(x+2)/(5) = (z-3)/( 1) , y=2` is

Options

A. parallel to x-axis
B. parallel to y-axis
C. parallel to z-axis
D. perpendicular to y-axis

Answer

To determine whether the given line \(\frac{x+2}{5} = \frac{z-3}{1}, y=2\) is parallel to the x, y, or z-axis, or perpendicular to the y-axis, we can follow these steps:

### Step 1: Rewrite the line equations
The line is given in the symmetric form. We can express it in a more standard vector form.

The equation can be rewritten as:
\[ \frac{x + 2}{5} = \frac{z - 3}{1} = y - 2 \]
Let’s denote this common ratio as \(k\). Thus, we can express \(x\), \(y\), and \(z\) in terms of \(k\):
\[ x + 2 = 5k \quad \Rightarrow \quad x = 5k - 2 \]
\[ y = 2 \quad \Rightarrow \quad y = 2 \]
\[ z - 3 = k \quad \Rightarrow \quad z = k + 3 \]

### Step 2: Identify direction ratios
From the equations, we can identify the direction ratios of the line. The direction ratios can be derived from the coefficients of \(k\):
- For \(x\): The coefficient is \(5\).
- For \(y\): The coefficient is \(0\) (since \(y\) is constant).
- For \(z\): The coefficient is \(1\).

Thus, the direction ratios of the line are \(5, 0, 1\).

### Step 3: Determine the relationship with axes
To check if the line is parallel or perpendicular to the axes, we can analyze the direction ratios:
- The line has a non-zero direction ratio for \(x\) (which is \(5\)) and for \(z\) (which is \(1\)), but the direction ratio for \(y\) is \(0\).

### Step 4: Check perpendicularity to the y-axis
Since the direction ratio for \(y\) is \(0\), it indicates that the line does not change in the \(y\) direction. Therefore, it is perpendicular to the y-axis.

### Conclusion
The line \(\frac{x+2}{5} = \frac{z-3}{1}, y=2\) is perpendicular to the y-axis.

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