Distance of the point `(x_(1),y_(1),z_(1))` from the line
`(x-x_(2))/1=(y-y_(2))/m=(z-z_(2))/n`
where l,m,n are direction cosines of the line, is

Options

A. `[(x_(1)-x_(2))^(z)+(y_(2)-y_(1))^(2)+(z_(1)+z_(2))^(2)-{l(x_(1)-x_(2))+m(y_(1)-y_(2))+n(z_(1)-z_(2))}^(2)]^(2//1)`
B. `sqrt((x_(2)-x_(1))^(2)+(y_(2)_y_(1))^(2)+(z_(2)-z_(1))^(2))`
C. `sqrt((x_(2)-x_(1))l+(y_(2)-y_(1))m+(z_(2)-z_(1))n)`
D. none of these

Answer

To find the distance of the point \( (x_1, y_1, z_1) \) from the line given by the equation

\[ \frac{x - x_2}{1} = \frac{y - y_2}{m} = \frac{z - z_2}{n} \]

where \( l, m, n \) are the direction cosines of the line, we can follow these steps:

### Step 1: Understand the Line Equation
The line can be expressed in parametric form as:
- \( x = x_2 + t \)
- \( y = y_2 + mt \)
- \( z = z_2 + nt \)

where \( t \) is a parameter.

### Step 2: Find a Point on the Line
Let \( P(t) \) be a point on the line corresponding to the parameter \( t \):
\[ P(t) = (x_2 + t, y_2 + mt, z_2 + nt) \]

### Step 3: Find the Distance Formula
The distance \( D \) between the point \( (x_1, y_1, z_1) \) and the point \( P(t) \) on the line is given by the distance formula:
\[ D = \sqrt{(x_1 - (x_2 + t))^2 + (y_1 - (y_2 + mt))^2 + (z_1 - (z_2 + nt))^2} \]

### Step 4: Simplify the Distance Expression
Substituting \( P(t) \) into the distance formula, we have:
\[ D = \sqrt{(x_1 - x_2 - t)^2 + (y_1 - y_2 - mt)^2 + (z_1 - z_2 - nt)^2} \]

### Step 5: Minimize the Distance
To find the minimum distance, we need to minimize \( D^2 \) (since minimizing \( D^2 \) will also minimize \( D \)):
\[ D^2 = (x_1 - x_2 - t)^2 + (y_1 - y_2 - mt)^2 + (z_1 - z_2 - nt)^2 \]

### Step 6: Differentiate with Respect to \( t \)
To find the value of \( t \) that minimizes \( D^2 \), we differentiate \( D^2 \) with respect to \( t \) and set the derivative to zero:
\[ \frac{d(D^2)}{dt} = 0 \]

### Step 7: Solve for \( t \)
After differentiating and simplifying, we can solve for \( t \) to find the specific point on the line that is closest to the point \( (x_1, y_1, z_1) \).

### Step 8: Substitute \( t \) Back to Find the Minimum Distance
Once we have the value of \( t \), we substitute it back into the distance formula \( D \) to find the minimum distance from the point to the line.

### Final Result
The final expression for the distance \( D \) can be derived from the above steps, leading to the conclusion that the distance of the point \( (x_1, y_1, z_1) \) from the line is given by the formula:
\[ D = \frac{|(x_1 - x_2)(m) + (y_1 - y_2)(-1) + (z_1 - z_2)(n)|}{\sqrt{1 + m^2 + n^2}} \]