From the point `P(3,-1,11)` a perpendicular is drawn on the line L given by the equation `x/2=(y-2)/3=(z-3)/4`. Let Q be the foot of the perpendicular
What is the length of the line segment PQ?

Options

A. A. `sqrt(47)` unit
B. B. 7 unit
C. C. `sqrt(53)` unit
D. D. 8 unit

Answer

To find the length of the line segment \( PQ \) where \( P(3, -1, 11) \) is a point and \( Q \) is the foot of the perpendicular from \( P \) to the line \( L \) given by the equation \( \frac{x}{2} = \frac{y-2}{3} = \frac{z-3}{4} \), we can follow these steps:

### Step 1: Parameterize the Line \( L \)
The line \( L \) can be parameterized using a parameter \( \lambda \):
- From the equation \( \frac{x}{2} = \frac{y-2}{3} = \frac{z-3}{4} = \lambda \), we can express the coordinates of any point \( Q \) on the line as:
\[ x = 2\lambda, \quad y = 3\lambda + 2, \quad z = 4\lambda + 3 \]

### Step 2: Find the Directional Cosines of \( PQ \)
Let \( Q \) have coordinates \( (2\lambda, 3\lambda + 2, 4\lambda + 3) \). The directional cosines of the line segment \( PQ \) can be expressed as:
- \( PQ = (2\lambda - 3, 3\lambda + 2 + 1, 4\lambda + 3 - 11) \)
- Simplifying gives:
\[ PQ = (2\lambda - 3, 3\lambda + 3, 4\lambda - 8) \]

### Step 3: Use the Perpendicular Condition
Since \( PQ \) is perpendicular to the line \( L \), the dot product of the directional cosines of \( PQ \) and the direction ratios of line \( L \) must equal zero. The direction ratios of line \( L \) are \( (2, 3, 4) \).

Setting up the equation:
\[ (2\lambda - 3) \cdot 2 + (3\lambda + 3) \cdot 3 + (4\lambda - 8) \cdot 4 = 0 \]

### Step 4: Solve for \( \lambda \)
Expanding the equation:
\[ 4\lambda - 6 + 9\lambda + 9 + 16\lambda - 32 = 0 \]
Combining like terms:
\[ (4\lambda + 9\lambda + 16\lambda) + (-6 + 9 - 32) = 0 \]
This simplifies to:
\[ 29\lambda - 29 = 0 \]
Thus, we find:
\[ \lambda = 1 \]

### Step 5: Find the Coordinates of \( Q \)
Substituting \( \lambda = 1 \) back into the parameterization:
\[ Q = (2 \cdot 1, 3 \cdot 1 + 2, 4 \cdot 1 + 3) = (2, 5, 7) \]

### Step 6: Calculate the Length of Segment \( PQ \)
Now, we can calculate the length of segment \( PQ \):
\[ PQ = \sqrt{(2 - 3)^2 + (5 - (-1))^2 + (7 - 11)^2} \]
Calculating each term:
\[ = \sqrt{(-1)^2 + (5 + 1)^2 + (-4)^2} = \sqrt{1 + 6^2 + 16} = \sqrt{1 + 36 + 16} = \sqrt{53} \]

### Conclusion
Thus, the length of the line segment \( PQ \) is:
\[ \text{Length of } PQ = \sqrt{53} \text{ units} \]