The co-ordinates of the foot of the perpendicular from the point `(3,-1,11)` on the line `(x)/(2)=(y-2)/(3)=(z-3)/(4)` are

Options

A. `(2,5,7)`
B. `(-2,-1,-1)`
C. `(0,2,3)`
D. `(2,3,4)`

Answer

To find the coordinates of the foot of the perpendicular from the point \( (3, -1, 11) \) on the line given by the equation \( \frac{x}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \), we will follow these steps:

### Step 1: Parameterize the Line
We start by expressing the line in parametric form. Let \( \lambda \) be the parameter such that:
\[ \frac{x}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} = \lambda \]
From this, we can express \( x, y, z \) in terms of \( \lambda \):
\[ x = 2\lambda, \quad y = 3\lambda + 2, \quad z = 4\lambda + 3 \]

### Step 2: Identify the Point and Direction Ratios
The point from which we are dropping the perpendicular is \( P(3, -1, 11) \). The coordinates of the foot of the perpendicular \( F \) on the line can be expressed as \( F(2\lambda, 3\lambda + 2, 4\lambda + 3) \).

### Step 3: Find Direction Ratios of the Perpendicular
The direction ratios of the line joining the point \( P \) and the foot \( F \) are given by:
\[ \text{Direction Ratios} = (2\lambda - 3, (3\lambda + 2) - (-1), (4\lambda + 3) - 11) \]
This simplifies to:
\[ (2\lambda - 3, 3\lambda + 3, 4\lambda - 8) \]

### Step 4: Find Direction Ratios of the Line
The direction ratios of the line given by the equation are \( (2, 3, 4) \).

### Step 5: Use the Perpendicular Condition
For two lines to be perpendicular, the dot product of their direction ratios must be zero:
\[ (2\lambda - 3) \cdot 2 + (3\lambda + 3) \cdot 3 + (4\lambda - 8) \cdot 4 = 0 \]
Expanding this gives:
\[ 4\lambda - 6 + 9\lambda + 9 + 16\lambda - 32 = 0 \]
Combining like terms:
\[ (4\lambda + 9\lambda + 16\lambda) + (-6 + 9 - 32) = 0 \]
This simplifies to:
\[ 29\lambda - 29 = 0 \]

### Step 6: Solve for \( \lambda \)
Setting the equation to zero:
\[ 29\lambda = 29 \implies \lambda = 1 \]

### Step 7: Find the Coordinates of the Foot of the Perpendicular
Substituting \( \lambda = 1 \) back into the parametric equations:
\[ x = 2(1) = 2, \quad y = 3(1) + 2 = 5, \quad z = 4(1) + 3 = 7 \]
Thus, the coordinates of the foot of the perpendicular are:
\[ (2, 5, 7) \]

### Final Answer
The coordinates of the foot of the perpendicular from the point \( (3, -1, 11) \) on the line are \( (2, 5, 7) \).

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