The distance of the plane `x+2y-z=2` from the point `(2, -1, 3)`, measured in the direction with the direction ratios (2, 2,1) is

Options

A. 1 units
B. 2 units
C. 3 units
D. 4 units

Answer

To find the distance of the plane \( x + 2y - z = 2 \) from the point \( (2, -1, 3) \) measured in the direction of the direction ratios \( (2, 2, 1) \), we can follow these steps:

### Step 1: Identify the point and the plane
We have the point \( A(2, -1, 3) \) and the plane equation \( x + 2y - z = 2 \).

### Step 2: Define the direction ratios
The direction ratios are given as \( (2, 2, 1) \). We can express a point \( B \) in the direction of these ratios from point \( A \) using a parameter \( \lambda \):
\[ B(x, y, z) = (2 + 2\lambda, -1 + 2\lambda, 3 + \lambda) \]

### Step 3: Substitute the coordinates of point B into the plane equation
Since point \( B \) lies on the plane, we substitute its coordinates into the plane equation:
\[ (2 + 2\lambda) + 2(-1 + 2\lambda) - (3 + \lambda) = 2 \]
Expanding this:
\[ 2 + 2\lambda - 2 + 4\lambda - 3 - \lambda = 2 \]
Simplifying:
\[ (2\lambda + 4\lambda - \lambda) + (2 - 2 - 3) = 2 \]
\[ 5\lambda - 3 = 2 \]
\[ 5\lambda = 5 \implies \lambda = 1 \]

### Step 4: Find the coordinates of point B
Substituting \( \lambda = 1 \) back into the coordinates of point \( B \):
\[ B = (2 + 2 \cdot 1, -1 + 2 \cdot 1, 3 + 1) = (4, 1, 4) \]

### Step 5: Calculate the distance between points A and B
Now, we can find the distance \( AB \) using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of points \( A(2, -1, 3) \) and \( B(4, 1, 4) \):
\[ AB = \sqrt{(4 - 2)^2 + (1 - (-1))^2 + (4 - 3)^2} \]
Calculating each term:
\[ = \sqrt{(2)^2 + (2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Final Answer
The distance of the plane from the point \( (2, -1, 3) \) measured in the direction of \( (2, 2, 1) \) is \( 3 \) units.

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