If the perpendicular distance of the plane 2x+3y-z= k from the origin is ` sqrt(14)` units then k=……..

Options

A. 14
B. 196
C. ` 2 sqrt (14)`
D. ` (sqrt(14))/(2)`

Answer

To find the value of \( k \) such that the perpendicular distance of the plane \( 2x + 3y - z = k \) from the origin is \( \sqrt{14} \) units, we can use the formula for the distance \( D \) from a point \( (x_1, y_1, z_1) \) to the plane \( Ax + By + Cz + D = 0 \):

\[ D = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

### Step-by-Step Solution:

1. **Identify the coefficients**: For the plane equation \( 2x + 3y - z = k \), we can rewrite it in the standard form:
\[ 2x + 3y - z - k = 0 \]
Here, \( A = 2 \), \( B = 3 \), \( C = -1 \), and \( D = -k \).

2. **Substitute the coordinates of the origin**: The coordinates of the origin are \( (0, 0, 0) \). Substitute \( x_1 = 0 \), \( y_1 = 0 \), \( z_1 = 0 \) into the distance formula:
\[ D = \frac{|2(0) + 3(0) - (0) - k|}{\sqrt{2^2 + 3^2 + (-1)^2}} \]
This simplifies to:
\[ D = \frac{| -k |}{\sqrt{4 + 9 + 1}} = \frac{|k|}{\sqrt{14}} \]

3. **Set the distance equal to \( \sqrt{14} \)**: According to the problem, this distance is given as \( \sqrt{14} \):
\[ \frac{|k|}{\sqrt{14}} = \sqrt{14} \]

4. **Cross-multiply to solve for \( |k| \)**:
\[ |k| = \sqrt{14} \cdot \sqrt{14} = 14 \]

5. **Determine the values of \( k \)**: Since \( |k| = 14 \), we have two possible values for \( k \):
\[ k = 14 \quad \text{or} \quad k = -14 \]

### Final Answer:
Thus, the possible values of \( k \) are \( k = 14 \) or \( k = -14 \).

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