Find the equations of the perpendicular drawn from the point P(2,4,-1) to the line:
`(x + 5)/(1) = (y + 3)/(4) = (z-6)/(-9)`.
Also, write down the co-ordinates of the foot of the perpendicular from P to the line.

Answer

To find the equations of the perpendicular drawn from the point \( P(2, 4, -1) \) to the line given by

\[ \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}, \]

we will follow these steps:

### Step 1: Identify a point on the line and the direction ratios of the line.
The line can be expressed in parametric form. From the equation, we can identify:

- A point on the line \( A(-5, -3, 6) \) (when \( \lambda = 0 \)).
- The direction ratios of the line \( \mathbf{b} = (1, 4, -9) \).

### Step 2: Set up the coordinates of a point on the line in terms of \( \lambda \).
Using the parameter \( \lambda \), we can express the coordinates of any point \( B \) on the line as:

\[ B(\lambda) = (-5 + \lambda, -3 + 4\lambda, 6 - 9\lambda). \]

### Step 3: Find the vector \( \overrightarrow{PB} \).
The vector from point \( P(2, 4, -1) \) to point \( B(\lambda) \) is given by:

\[ \overrightarrow{PB} = B(\lambda) - P = \left((-5 + \lambda) - 2, (-3 + 4\lambda) - 4, (6 - 9\lambda) - (-1)\right). \]

This simplifies to:

\[ \overrightarrow{PB} = (\lambda - 7, 4\lambda - 7, 7 - 9\lambda). \]

### Step 4: Set up the perpendicularity condition.
For the line \( \overrightarrow{PB} \) to be perpendicular to the direction vector of the line \( \mathbf{b} = (1, 4, -9) \), the dot product must equal zero:

\[ \overrightarrow{PB} \cdot \mathbf{b} = 0. \]

Calculating the dot product:

\[ (\lambda - 7) \cdot 1 + (4\lambda - 7) \cdot 4 + (7 - 9\lambda) \cdot (-9) = 0. \]

Expanding this:

\[ \lambda - 7 + 16\lambda - 28 - 63 + 81\lambda = 0. \]

Combining like terms gives:

\[ (1 + 16 + 81)\lambda - 7 - 28 - 63 = 0, \]
\[ 98\lambda - 98 = 0. \]

### Step 5: Solve for \( \lambda \).
From the equation \( 98\lambda - 98 = 0 \), we find:

\[ \lambda = 1. \]

### Step 6: Find the coordinates of the foot of the perpendicular.
Substituting \( \lambda = 1 \) back into the parametric equations for the line:

\[ B(1) = (-5 + 1, -3 + 4 \cdot 1, 6 - 9 \cdot 1) = (-4, 1, -3). \]

Thus, the coordinates of the foot of the perpendicular from \( P \) to the line are \( B(-4, 1, -3) \).

### Step 7: Write the equations of the perpendicular.
The direction ratios of the perpendicular from \( P(2, 4, -1) \) to \( B(-4, 1, -3) \) can be calculated as:

\[ \overrightarrow{PB} = (-4 - 2, 1 - 4, -3 + 1) = (-6, -3, -2). \]

The equations of the line can be expressed in symmetric form as:

\[ \frac{x - 2}{-6} = \frac{y - 4}{-3} = \frac{z + 1}{-2}. \]

### Final Result:
The equations of the perpendicular from point \( P(2, 4, -1) \) to the line are:

\[ \frac{x - 2}{-6} = \frac{y - 4}{-3} = \frac{z + 1}{-2}, \]

and the coordinates of the foot of the perpendicular are \( B(-4, 1, -3) \).