If the equation of a plane is `lx + my + nz = p` which is in the normal form, then which one of the following is not true?

Options

A. l,m and n are the direction cosines of the normal to the plane
B. `p` is the length of the perpendicular from the origin to the plane
C. the plane passes through theorigin for all values of ``
D. `l^(2)+m^(2)+n^(2)=1`

Explanation

To solve the problem, we need to analyze the equation of the plane given in the normal form, which is \( lx + my + nz = p \). We will evaluate the statements provided in the options to determine which one is not true.

### Step 1: Understand the Normal Form of the Plane
The equation of the plane in normal form is given as:
\[ lx + my + nz = p \]
Here, \( (l, m, n) \) represents the direction ratios of the normal to the plane, and \( p \) is a constant.

### Step 2: Identify Direction Ratios vs. Direction Cosines
Direction ratios are proportional to the direction cosines but are not necessarily normalized. The direction cosines \( (l', m', n') \) satisfy the condition:
\[ l'^2 + m'^2 + n'^2 = 1 \]
However, for direction ratios \( (l, m, n) \), this condition does not hold true unless they are normalized.

### Step 3: Evaluate the Statement about Direction Ratios
Given that \( l, m, n \) are direction ratios of the normal, we can conclude that:
- The statement claiming \( l, m, n \) are direction cosines is **not true**.

### Step 4: Analyze the Perpendicular Distance from the Origin
To find the perpendicular distance from the origin to the plane, we can substitute \( (0, 0, 0) \) into the plane equation:
\[ l(0) + m(0) + n(0) = p \implies 0 = p \]
This implies that for the plane to pass through the origin, \( p \) must be equal to 0.

### Step 5: Conclusion on the Options
1. The statement that \( l, m, n \) are direction cosines is **not true**.
2. The statement that \( p \) is the length of the perpendicular from the origin to the plane is **true** only if \( p = 0 \).
3. The statement that the plane passes through the origin for all values of \( p \) is **not true**; it only passes through the origin when \( p = 0 \).

Thus, the option that is **not true** is the one that states that \( l, m, n \) are direction cosines.

### Summary of Findings
- **Not True**: \( l, m, n \) are direction cosines.
- **True**: \( p \) is the length of the perpendicular from the origin to the plane (only if \( p = 0 \)).
- **Not True**: The plane passes through the origin for all values of \( p \).