To find the foot of the perpendicular from the point \( P(2, 4, -1) \) to the given line, we will follow these steps:
### Step 1: Parametrize the Line
The line is given in the symmetric form:
\[
\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}
\]
Let \( \lambda \) be the parameter. Then we can express the coordinates \( (x, y, z) \) in terms of \( \lambda \):
\[
x = 1\lambda - 5 \\
y = 4\lambda - 3 \\
z = -9\lambda + 6
\]
### Step 2: Define the Point on the Line
Let \( Q \) be the point on the line corresponding to the parameter \( \lambda \):
\[
Q(\lambda) = (1\lambda - 5, 4\lambda - 3, -9\lambda + 6)
\]
### Step 3: Find the Direction Ratios of PQ
The direction ratios of the line segment \( PQ \) can be found by subtracting the coordinates of \( P \) from those of \( Q \):
\[
PQ = Q - P = \left( (1\lambda - 5) - 2, (4\lambda - 3) - 4, (-9\lambda + 6) - (-1) \right)
\]
This simplifies to:
\[
PQ = (1\lambda - 7, 4\lambda - 7, -9\lambda + 7)
\]
### Step 4: Direction Ratios of the Line
The direction ratios of the line are given by the coefficients from the symmetric form:
\[
d = (1, 4, -9)
\]
### Step 5: Use the Perpendicular Condition
For \( PQ \) to be perpendicular to the line, the dot product of the direction ratios must equal zero:
\[
(1\lambda - 7) \cdot 1 + (4\lambda - 7) \cdot 4 + (-9\lambda + 7) \cdot (-9) = 0
\]
Expanding this:
\[
(1\lambda - 7) + 4(4\lambda - 7) + 9(9\lambda - 7) = 0
\]
This simplifies to:
\[
\lambda - 7 + 16\lambda - 28 + 81\lambda - 63 = 0
\]
Combining like terms:
\[
98\lambda - 98 = 0
\]
So,
\[
\lambda = 1
\]
### Step 6: Substitute \( \lambda \) Back to Find Coordinates of Q
Now substitute \( \lambda = 1 \) back into the equations for \( Q \):
\[
Q(1) = (1 \cdot 1 - 5, 4 \cdot 1 - 3, -9 \cdot 1 + 6)
\]
Calculating each coordinate:
\[
Q(1) = (1 - 5, 4 - 3, -9 + 6) = (-4, 1, -3)
\]
### Final Answer
Thus, the foot of the perpendicular from the point \( (2, 4, -1) \) to the line is:
\[
\boxed{(-4, 1, -3)}
\]
