O is origin and P is a point at a distance of 7 units from origin if the direction ratios of OP are `lt 3 ,-2, 6 gt`, then the coordinates of P are (i) `(3,-2,6)` (ii) `(21,-14,42)` (iii) `(3/7,-2/7,6/7)` (iv) none of these

Options

A. `(3, -2,6)`
B. `(21,-14,42)`
C. `((3)/(7),-(2)/(7),(6)/(7) )`
D. none of these

Answer

To find the coordinates of point P given that O is the origin and P is at a distance of 7 units from the origin with direction ratios of OP as (3, -2, 6), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have the origin O at (0, 0, 0).
- The point P is at a distance of 7 units from O.
- The direction ratios of OP are given as (3, -2, 6).

2. **Using the Distance Formula**:
- The distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (0, 0, 0) \) is given by:
\[ d = \sqrt{(x_1 - 0)^2 + (y_1 - 0)^2 + (z_1 - 0)^2} \]
- Since the distance \( d \) is 7, we can write:
\[ \sqrt{x_1^2 + y_1^2 + z_1^2} = 7 \]
- Squaring both sides gives:
\[ x_1^2 + y_1^2 + z_1^2 = 49 \quad \text{(Equation 1)} \]

3. **Using Direction Ratios**:
- The direction ratios (3, -2, 6) can be used to express the coordinates of point P in terms of a scalar multiple \( k \):
\[ x_1 = 3k, \quad y_1 = -2k, \quad z_1 = 6k \]

4. **Substituting into the Distance Equation**:
- Substitute \( x_1, y_1, z_1 \) into Equation 1:
\[ (3k)^2 + (-2k)^2 + (6k)^2 = 49 \]
- Simplifying this gives:
\[ 9k^2 + 4k^2 + 36k^2 = 49 \]
\[ 49k^2 = 49 \]
- Dividing both sides by 49:
\[ k^2 = 1 \quad \Rightarrow \quad k = 1 \text{ or } k = -1 \]

5. **Finding Coordinates for k = 1**:
- If \( k = 1 \):
\[ x_1 = 3(1) = 3, \quad y_1 = -2(1) = -2, \quad z_1 = 6(1) = 6 \]
- Thus, the coordinates of point P are \( (3, -2, 6) \).

6. **Finding Coordinates for k = -1**:
- If \( k = -1 \):
\[ x_1 = 3(-1) = -3, \quad y_1 = -2(-1) = 2, \quad z_1 = 6(-1) = -6 \]
- Thus, the coordinates of point P would be \( (-3, 2, -6) \), which is not in the options.

7. **Conclusion**:
- The coordinates of point P that satisfy the conditions of the problem are \( (3, -2, 6) \).
- Therefore, the correct option is (i) \( (3, -2, 6) \).