The foot of the perpendicular form `(2,4,-1)` to the line `x +5 =(1)/(4) (y+3) = -(1)/(9) (z - 6)` is

Options

A. `(-4,1,-3)`
B. `(4,-1,-3)`
C. `(-4,-1,3)`
D. `(-4,-1,3)`

Explanation

To find the foot of the perpendicular from the point \( P(2, 4, -1) \) to the line given by the equations \( x + 5 = \frac{1}{4}(y + 3) = -\frac{1}{9}(z - 6) \), we can follow these steps:

### Step 1: Parametrize the Line
The line can be expressed in parametric form. Let \( \lambda \) be the parameter. We can rewrite the equations as follows:

\[ x + 5 = \lambda \implies x = \lambda - 5 \]
\[ \frac{1}{4}(y + 3) = \lambda \implies y = 4\lambda - 3 \]
\[ -\frac{1}{9}(z - 6) = \lambda \implies z = -9\lambda + 6 \]

Thus, any point \( Q \) on the line can be represented as:
\[ Q(\lambda - 5, 4\lambda - 3, -9\lambda + 6) \]

### Step 2: Find Direction Ratios of PQ
Let \( Q = (x_Q, y_Q, z_Q) = (\lambda - 5, 4\lambda - 3, -9\lambda + 6) \). The direction ratios of the line from point \( P(2, 4, -1) \) to point \( Q \) are given by:
\[ PQ = (x_Q - 2, y_Q - 4, z_Q + 1) = ((\lambda - 5) - 2, (4\lambda - 3) - 4, (-9\lambda + 6) + 1) \]
This simplifies to:
\[ PQ = (\lambda - 7, 4\lambda - 7, -9\lambda + 7) \]

### Step 3: Find Direction Ratios of the Line
The direction ratios of the line can be derived from its parametric equations:
- From \( x \): the coefficient is \( 1 \)
- From \( y \): the coefficient is \( 4 \)
- From \( z \): the coefficient is \( -9 \)

Thus, the direction ratios of the line are \( (1, 4, -9) \).

### Step 4: Set Up the Dot Product Equation
Since \( PQ \) is perpendicular to the line, their dot product must equal zero:
\[ (\lambda - 7) \cdot 1 + (4\lambda - 7) \cdot 4 + (-9\lambda + 7) \cdot (-9) = 0 \]

Expanding this gives:
\[ \lambda - 7 + 16\lambda - 28 + 81\lambda - 63 = 0 \]
Combining like terms:
\[ 98\lambda - 98 = 0 \]

### Step 5: Solve for \( \lambda \)
Solving for \( \lambda \):
\[ 98\lambda = 98 \implies \lambda = 1 \]

### Step 6: Find Coordinates of the Foot of the Perpendicular
Substituting \( \lambda = 1 \) back into the parametric equations of the line:
\[ x = 1 - 5 = -4 \]
\[ y = 4(1) - 3 = 1 \]
\[ z = -9(1) + 6 = -3 \]

Thus, the coordinates of the foot of the perpendicular \( Q \) are:
\[ Q(-4, 1, -3) \]

### Final Answer
The foot of the perpendicular from the point \( (2, 4, -1) \) to the line is \( Q(-4, 1, -3) \).

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