L is the foot of perpendicular drawn from a point P(3,4,5) on x-axis. The coordinates of L are

Options

A. (3,0,0)
B. (0,4,0)
C. (0,0,5)
D. (0, 4, 5)

Answer

To find the coordinates of point L, which is the foot of the perpendicular drawn from point P(3, 4, 5) to the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the position of point L**: Since L is the foot of the perpendicular from point P to the x-axis, its y and z coordinates will be 0. Therefore, we can denote the coordinates of point L as \( L(\lambda, 0, 0) \), where \( \lambda \) is the x-coordinate we need to find.

2. **Identify the coordinates of point P**: The coordinates of point P are given as \( P(3, 4, 5) \).

3. **Determine the vectors**:
- The vector from P to L can be represented as \( \overrightarrow{PL} = L - P = (\lambda, 0, 0) - (3, 4, 5) = (\lambda - 3, -4, -5) \).
- The vector from the origin O to L is \( \overrightarrow{OL} = L - O = (\lambda, 0, 0) - (0, 0, 0) = (\lambda, 0, 0) \).

4. **Use the dot product for perpendicularity**: Since the line PL is perpendicular to the x-axis (represented by vector OL), the dot product of these two vectors should equal zero:
\[ \overrightarrow{PL} \cdot \overrightarrow{OL} = 0 \]
This gives us:
\[ (\lambda - 3, -4, -5) \cdot (\lambda, 0, 0) = 0 \]

5. **Calculate the dot product**:
\[ (\lambda - 3) \cdot \lambda + (-4) \cdot 0 + (-5) \cdot 0 = 0 \]
Simplifying this, we have:
\[ \lambda^2 - 3\lambda = 0 \]

6. **Factor the equation**:
\[ \lambda(\lambda - 3) = 0 \]
This gives us two possible solutions: \( \lambda = 0 \) or \( \lambda = 3 \).

7. **Determine the valid solution**:
- If \( \lambda = 0 \), then point L would be at the origin (0, 0, 0), which is not the foot of the perpendicular from P.
- Therefore, we take \( \lambda = 3 \).

8. **Write the coordinates of point L**:
The coordinates of point L are \( L(3, 0, 0) \).

### Final Answer:
The coordinates of L are \( (3, 0, 0) \).