The perpendicular distance of the origin from the plane x-3y+4z=6 is ….

Options

A. ` 6`
B. ` (6)/(sqrt(26))`
C. ` 36`
D. `(1)/(sqrt(26))`

Answer

To find the perpendicular distance of the origin from the plane given by the equation \( x - 3y + 4z = 6 \), we can use the formula for the distance \( D \) from a point \( (x_1, y_1, z_1) \) to the plane \( ax + by + cz + d = 0 \).

### Step-by-Step Solution:

1. **Identify the Plane Equation**:
The given plane equation is \( x - 3y + 4z = 6 \). We can rewrite it in the standard form:
\[ x - 3y + 4z - 6 = 0 \]
Here, \( a = 1 \), \( b = -3 \), \( c = 4 \), and \( d = 6 \).

2. **Identify the Point**:
We need to find the distance from the origin, which is the point \( (0, 0, 0) \). Thus, \( x_1 = 0 \), \( y_1 = 0 \), and \( z_1 = 0 \).

3. **Substitute into the Distance Formula**:
The formula for the distance \( D \) from a point \( (x_1, y_1, z_1) \) to the plane \( ax + by + cz + d = 0 \) is given by:
\[ D = \frac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}} \]
Substituting the values:
\[ D = \frac{|1(0) + (-3)(0) + 4(0) - 6|}{\sqrt{1^2 + (-3)^2 + 4^2}} \]

4. **Calculate the Numerator**:
The numerator simplifies to:
\[ |0 + 0 + 0 - 6| = |-6| = 6 \]

5. **Calculate the Denominator**:
Now calculate the denominator:
\[ \sqrt{1^2 + (-3)^2 + 4^2} = \sqrt{1 + 9 + 16} = \sqrt{26} \]

6. **Final Calculation of Distance**:
Now, substituting back into the distance formula:
\[ D = \frac{6}{\sqrt{26}} \]

### Conclusion:
The perpendicular distance of the origin from the plane \( x - 3y + 4z = 6 \) is:
\[ D = \frac{6}{\sqrt{26}} \]