What is the length of the perpendicular from the origin to the plane `ax+by+sqrt(2abz)=1` ?

Options

A. `1//(ab)`
B. `1//(a+b)`
C. `a+b`
D. `ab`

Answer

To find the length of the perpendicular from the origin to the plane given by the equation \( ax + by + \sqrt{2ab}z = 1 \), we can use the formula for the distance \( D \) from a point to a plane defined by the equation \( Ax + By + Cz + D = 0 \).

### Step-by-Step Solution:

1. **Identify the Plane Equation**:
The given plane equation is \( ax + by + \sqrt{2ab}z = 1 \). We can rewrite this in the standard form:
\[ ax + by + \sqrt{2ab}z - 1 = 0 \]
Here, \( A = a \), \( B = b \), \( C = \sqrt{2ab} \), and \( D = -1 \).

2. **Identify the Point**:
The point from which we want to find the perpendicular distance is the origin, which has coordinates \( (0, 0, 0) \).

3. **Use the Distance Formula**:
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 \( (x_1, y_1, z_1) = (0, 0, 0) \):
\[ D = \frac{|A(0) + B(0) + C(0) + D|}{\sqrt{A^2 + B^2 + C^2}} = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}} \]

4. **Calculate \( |D| \)**:
We have \( D = -1 \), thus:
\[ |D| = |-1| = 1 \]

5. **Calculate \( A^2 + B^2 + C^2 \)**:
Now, we calculate \( A^2 + B^2 + C^2 \):
\[ A^2 = a^2, \quad B^2 = b^2, \quad C^2 = (\sqrt{2ab})^2 = 2ab \]
Therefore:
\[ A^2 + B^2 + C^2 = a^2 + b^2 + 2ab = (a + b)^2 \]

6. **Final Distance Calculation**:
Substitute back into the distance formula:
\[ D = \frac{1}{\sqrt{(a + b)^2}} = \frac{1}{|a + b|} \]

### Final Answer:
The length of the perpendicular from the origin to the plane \( ax + by + \sqrt{2ab}z = 1 \) is:
\[ D = \frac{1}{|a + b|} \]