`A(-2, 2, 3) and B(13, -3, 13)` and L is a line through A.
Q. Equation of a line L, perpendicular to the line AB is

Options

A. `(x+2)/(15)=(y-2)/(-5)=(z-3)/(10)`
B. `(x-2)/(3)=(y+2)/(13)=(z+3)/(2)`
C. `(x+2)/(3)=(y-2)/(13)=(z-3)/(2)`
D. `(x-2)/(15)=(y+2)/(-5)=(z+3)/(10)`

Answer

To find the equation of the line \( L \) that passes through point \( A(-2, 2, 3) \) and is perpendicular to the line segment \( AB \) where \( B(13, -3, 13) \), we will follow these steps:

### Step 1: Find the direction ratios of line \( AB \)

The direction ratios (DRs) of line \( AB \) can be calculated using the coordinates of points \( A \) and \( B \). The formula for finding the direction ratios from two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is:

\[ \text{DRs} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]

Substituting the coordinates of points \( A \) and \( B \):

\[ \text{DRs of } AB = (13 - (-2), -3 - 2, 13 - 3) = (13 + 2, -5, 10) = (15, -5, 10) \]

### Step 2: Write the equation of line \( L \)

The equation of a line in three-dimensional space that passes through a point \( (x_1, y_1, z_1) \) and has direction ratios \( (a, b, c) \) is given by:

\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]

Since line \( L \) passes through point \( A(-2, 2, 3) \) and is perpendicular to line \( AB \), we need to find direction ratios \( (a, b, c) \) such that their dot product with the direction ratios of \( AB \) is zero (for perpendicularity).

### Step 3: Set up the perpendicularity condition

Let the direction ratios of line \( L \) be \( (a, b, c) \). The condition for perpendicularity is:

\[ 15a - 5b + 10c = 0 \]

### Step 4: Choose suitable direction ratios for line \( L \)

To find direction ratios \( (a, b, c) \) that satisfy the perpendicularity condition, we can choose values for \( a \) and \( b \) and solve for \( c \).

Let's choose \( a = 3 \) and \( b = 13 \):

\[ 15(3) - 5(13) + 10c = 0 \]
\[ 45 - 65 + 10c = 0 \]
\[ 10c = 20 \implies c = 2 \]

Thus, the direction ratios for line \( L \) can be \( (3, 13, 2) \).

### Step 5: Write the final equation of line \( L \)

Now substituting the values of \( a, b, c \) into the equation of line \( L \):

\[ \frac{x + 2}{3} = \frac{y - 2}{13} = \frac{z - 3}{2} \]

This is the equation of the line \( L \) that is perpendicular to line \( AB \) and passes through point \( A \).