What is the equation of the plane through z-axis and parallel to the line `(x-1)/(costheta)=(y+2)/(sintheta)=(z-3)/0`?

Options

A. A. `xcottheta+y=0`
B. B. `xtantheta-y=0`
C. C. `x+ycostheta=0`
D. D. `x-ytantheta=0`

Answer

To find the equation of the plane that passes through the z-axis and is parallel to the given line, we can follow these steps:

### Step 1: Identify the direction ratios of the line
The given line is represented as:
\[ \frac{x - 1}{\cos \theta} = \frac{y + 2}{\sin \theta} = \frac{z - 3}{0} \]
From this equation, we can extract the direction ratios of the line, which are:
\[ \text{Direction Ratios} = (\cos \theta, \sin \theta, 0) \]

### Step 2: Understand the condition of the plane
The plane must pass through the z-axis, which means it also passes through the origin (0, 0, 0). Therefore, the equation of the plane can be expressed in the general form:
\[ Ax + By + Cz = D \]
Since the plane passes through the origin, \(D = 0\).

### Step 3: Use the direction ratios to form the equation of the plane
Since the plane is parallel to the line, the direction ratios of the line will also be the coefficients of \(x\) and \(y\) in the equation of the plane. Thus, we can write:
\[ \cos \theta \cdot (x - 0) + \sin \theta \cdot (y - 0) + 0 \cdot (z - 0) = 0 \]
This simplifies to:
\[ \cos \theta \cdot x + \sin \theta \cdot y = 0 \]

### Step 4: Rearranging the equation
We can rearrange the equation:
\[ x \cos \theta + y \sin \theta = 0 \]

### Step 5: Further simplification
Dividing the entire equation by \(\sin \theta\) (assuming \(\sin \theta \neq 0\)), we get:
\[ x \cdot \frac{\cos \theta}{\sin \theta} + y = 0 \]
This can be rewritten as:
\[ x \cot \theta + y = 0 \]

### Final Equation
Thus, the equation of the plane is:
\[ x \cot \theta + y = 0 \]