A line passes through the point `A(2, 3, 5)` and is parallel to the vector `2hati-hatj+hatk`. If P is a point on this line such that `AP=2sqrt6`, then the coordinates of point P can be

Options

A. `(4, 2, 6)`
B. `(6, 1, 7)`
C. `(-2, 5, -3)`
D. `(2, 3, 5)`

Answer

To find the coordinates of point P on the line that passes through point A(2, 3, 5) and is parallel to the vector \( \mathbf{v} = 2\hat{i} - \hat{j} + \hat{k} \), we can follow these steps:

### Step 1: Identify the Direction Ratios
The direction ratios of the line are given by the components of the vector \( \mathbf{v} \). Thus, the direction ratios are:
- \( a = 2 \)
- \( b = -1 \)
- \( c = 1 \)

### Step 2: Write the Parametric Equations of the Line
Using the point A(2, 3, 5) and the direction ratios, we can write the parametric equations of the line:
- \( x = 2 + 2t \)
- \( y = 3 - t \)
- \( z = 5 + t \)

### Step 3: Distance Formula
We know that the distance \( AP \) is given as \( 2\sqrt{6} \). We can use the distance formula between points A(2, 3, 5) and P(x, y, z):
\[ AP = \sqrt{(x - 2)^2 + (y - 3)^2 + (z - 5)^2} \]
Substituting the parametric equations into the distance formula:
\[ AP = \sqrt{((2 + 2t) - 2)^2 + ((3 - t) - 3)^2 + ((5 + t) - 5)^2} \]
This simplifies to:
\[ AP = \sqrt{(2t)^2 + (-t)^2 + (t)^2} \]
\[ AP = \sqrt{4t^2 + t^2 + t^2} = \sqrt{6t^2} \]
\[ AP = \sqrt{6} |t| \]

### Step 4: Set Up the Equation
Since \( AP = 2\sqrt{6} \), we have:
\[ \sqrt{6} |t| = 2\sqrt{6} \]
Dividing both sides by \( \sqrt{6} \):
\[ |t| = 2 \]

### Step 5: Solve for t
This gives us two possible values for \( t \):
1. \( t = 2 \)
2. \( t = -2 \)

### Step 6: Find Coordinates of Point P
Now we substitute these values of \( t \) back into the parametric equations to find the coordinates of point P.

**For \( t = 2 \)**:
- \( x = 2 + 2(2) = 6 \)
- \( y = 3 - 2 = 1 \)
- \( z = 5 + 2 = 7 \)

So one possible point P is \( (6, 1, 7) \).

**For \( t = -2 \)**:
- \( x = 2 + 2(-2) = -2 \)
- \( y = 3 - (-2) = 5 \)
- \( z = 5 + (-2) = 3 \)

So another possible point P is \( (-2, 5, 3) \).

### Final Answer
The coordinates of point P can be \( (6, 1, 7) \) or \( (-2, 5, 3) \).