Equation of a line passing through point (1,2,3) and perpendicular to YZ-plane is

Options

A. `(x-1)/(1)=(y-1)/(1)=(z-3)/(0)`
B. `(x-1)/(1)=(y-2)/(0)=(z-3)/(0)`
C. `(x-1)/(0)=(y-2)/(1)=(z-3)/(1)`
D. `(x-1)/(0)=(y-2)/(0)=(z-3)/(1)`

Answer

To find the equation of a line passing through the point (1, 2, 3) and perpendicular to the YZ-plane, we can follow these steps:

### Step 1: Understand the Geometry
The YZ-plane consists of all points where the x-coordinate is zero. A line that is perpendicular to the YZ-plane must be parallel to the x-axis. This means that the direction ratios of the line will have a component only in the x-direction.

### Step 2: Identify the Direction Ratios
Since the line is perpendicular to the YZ-plane, the direction ratios can be defined as follows:
- The direction ratio along the x-axis (a) = 1 (since the line moves in the x-direction)
- The direction ratio along the y-axis (b) = 0 (no movement in the y-direction)
- The direction ratio along the z-axis (c) = 0 (no movement in the z-direction)

Thus, the direction ratios are (1, 0, 0).

### Step 3: Use the Point-Direction Form of the Line
The point-direction form of the line can be expressed as:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]
Here, (x1, y1, z1) is the point through which the line passes, and (a, b, c) are the direction ratios.

Substituting the values:
- (x1, y1, z1) = (1, 2, 3)
- (a, b, c) = (1, 0, 0)

The equation becomes:
\[ \frac{x - 1}{1} = \frac{y - 2}{0} = \frac{z - 3}{0} \]

### Step 4: Simplify the Equation
From the equation:
\[ \frac{x - 1}{1} = \frac{y - 2}{0} = \frac{z - 3}{0} \]
This implies:
- \(x - 1 = t\) (for some parameter t)
- \(y - 2 = 0\) (which means \(y = 2\))
- \(z - 3 = 0\) (which means \(z = 3\))

Thus, the final equation of the line is:
\[ x = 1 + t, \quad y = 2, \quad z = 3 \]

### Step 5: Write the Final Equation
In symmetric form, the equation of the line can be written as:
\[ x - 1 = 0, \quad y - 2 = 0, \quad z - 3 = 0 \]
This indicates that the line is indeed vertical along the x-axis, passing through the point (1, 2, 3).

### Final Answer
The equation of the line passing through the point (1, 2, 3) and perpendicular to the YZ-plane is:
\[ x = 1, \quad y = 2, \quad z = 3 \]