The foot of perpendicular of the point `(2,0,5)` on the line `(x+1)/(2)=(y-1)/(5)=(z+1)/(-1)` is `(alpha,beta,gamma)`.Then which of the following is NOT correct?

Options

A. `(gamma)/(alpha)=(5)/(8)`
B. `(beta)/(gamma)=-5`
C. `(alpha beta)/(gamma)=(4)/(15)`
D. `(alpha)/(beta)=-8`

Answer

To find the foot of the perpendicular from the point \( P(2, 0, 5) \) to the line given by the equations \( \frac{x + 1}{2} = \frac{y - 1}{5} = \frac{z + 1}{-1} \), we can follow these steps:

### Step 1: Parametrize the Line
The line can be parametrized using a parameter \( t \):
\[ x = 2t - 1, \quad y = 5t + 1, \quad z = -t - 1 \]

### Step 2: Write the Position Vector of the Point on the Line
The position vector \( \vec{F} \) of a point on the line is:
\[ \vec{F} = (2t - 1, 5t + 1, -t - 1) \]

### Step 3: Write the Vector from Point \( P \) to Point \( F \)
The vector \( \vec{PF} \) from point \( P(2, 0, 5) \) to point \( F \) is:
\[ \vec{PF} = \vec{F} - \vec{P} = (2t - 1 - 2, 5t + 1 - 0, -t - 1 - 5) = (2t - 3, 5t + 1, -t - 6) \]

### Step 4: Direction Vector of the Line
The direction vector \( \vec{A} \) of the line is:
\[ \vec{A} = (2, 5, -1) \]

### Step 5: Set Up the Dot Product Equation
Since \( \vec{PF} \) is perpendicular to \( \vec{A} \), we have:
\[ \vec{PF} \cdot \vec{A} = 0 \]
This gives:
\[ (2t - 3) \cdot 2 + (5t + 1) \cdot 5 + (-t - 6) \cdot (-1) = 0 \]
Expanding this:
\[ 4t - 6 + 25t + 5 + t + 6 = 0 \]
Combining like terms:
\[ 30t + 5 = 0 \]
Thus,
\[ t = -\frac{1}{6} \]

### Step 6: Substitute \( t \) Back into the Parametric Equations
Now substitute \( t = -\frac{1}{6} \) back into the parametric equations of the line to find the coordinates of point \( F \):
\[ x = 2\left(-\frac{1}{6}\right) - 1 = -\frac{1}{3} - 1 = -\frac{4}{3} \]
\[ y = 5\left(-\frac{1}{6}\right) + 1 = -\frac{5}{6} + 1 = \frac{1}{6} \]
\[ z = -\left(-\frac{1}{6}\right) - 1 = \frac{1}{6} - 1 = -\frac{5}{6} \]

Thus, the foot of the perpendicular \( F \) is:
\[ F\left(-\frac{4}{3}, \frac{1}{6}, -\frac{5}{6}\right) \]

### Step 7: Identify the Values of \( \alpha, \beta, \gamma \)
From the coordinates of \( F \):
\[ \alpha = -\frac{4}{3}, \quad \beta = \frac{1}{6}, \quad \gamma = -\frac{5}{6} \]

### Step 8: Analyze the Options
Now we need to check which of the given options is NOT correct based on the values of \( \alpha, \beta, \gamma \).