The length of perpendiculars from the point `P(1,2,6)` on the line
`L:(x-3)/(-2)=(y+1)/(1)=(z-5)/(2)`, is:

Options

A. `sqrt(2)`
B. `sqrt(3)`
C. 2
D. `sqrt(5)`

Answer

To find the length of the perpendicular from the point \( P(1, 2, 6) \) to the line given by the equation

\[ L: \frac{x-3}{-2} = \frac{y+1}{1} = \frac{z-5}{2}, \]

we will follow these steps:

### Step 1: Parametrize the Line
We can express the line in parametric form by letting

\[ \frac{x-3}{-2} = \frac{y+1}{1} = \frac{z-5}{2} = \lambda. \]

From this, we can derive the parametric equations for \( x, y, \) and \( z \):

\[ x = -2\lambda + 3, \]
\[ y = \lambda - 1, \]
\[ z = 2\lambda + 5. \]

### Step 2: Define the Point on the Line
Let \( Q \) be a point on the line corresponding to the parameter \( \lambda \):

\[ Q(-2\lambda + 3, \lambda - 1, 2\lambda + 5). \]

### Step 3: Find the Direction Ratios
The direction ratios of the line \( L \) can be extracted from the equation:

The direction ratios are given by the coefficients of \( \lambda \) in the parametric equations, which are \( (-2, 1, 2) \).

### Step 4: Find the Direction Ratios of \( PQ \)
The direction ratios of the line segment \( PQ \) can be calculated by subtracting the coordinates of \( P \) from those of \( Q \):

\[ PQ = Q - P = (-2\lambda + 3 - 1, \lambda - 1 - 2, 2\lambda + 5 - 6). \]

This simplifies to:

\[ PQ = (-2\lambda + 2, \lambda - 3, 2\lambda - 1). \]

### Step 5: Set Up the Dot Product Equation
Since \( PQ \) is perpendicular to \( L \), their dot product must equal zero:

\[ (-2\lambda + 2)(-2) + (\lambda - 3)(1) + (2\lambda - 1)(2) = 0. \]

Expanding this gives:

\[ 4\lambda - 4 + \lambda - 3 + 4\lambda - 2 = 0. \]

Combining like terms results in:

\[ 9\lambda - 9 = 0. \]

### Step 6: Solve for \( \lambda \)
Solving for \( \lambda \):

\[ 9\lambda = 9 \implies \lambda = 1. \]

### Step 7: Find Coordinates of Point \( Q \)
Substituting \( \lambda = 1 \) back into the equations for \( Q \):

\[ Q(-2(1) + 3, 1 - 1, 2(1) + 5) = Q(1, 0, 7). \]

### Step 8: Calculate the Distance \( PQ \)
Now we calculate the distance between points \( P(1, 2, 6) \) and \( Q(1, 0, 7) \) using the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}. \]

Substituting the coordinates:

\[ d = \sqrt{(1 - 1)^2 + (0 - 2)^2 + (7 - 6)^2} = \sqrt{0 + 4 + 1} = \sqrt{5}. \]

### Final Answer
The length of the perpendicular from the point \( P(1, 2, 6) \) to the line \( L \) is

\[ \sqrt{5} \text{ units}. \]

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