If a line with direction ration 2: 2: 1 intersects the lines `(x-7)/3=(y-5)/2=(z-3)/1 and (x-1) /2 =(y+1 )/4 = (z+1)/3` at A and B, respectively then AB

Options

A. `sqrt2`
B. 2
C. `sqrt3`
D. 3

Answer

To solve the problem, we need to find the distance \( AB \) between the points \( A \) and \( B \) where a line with direction ratios \( 2:2:1 \) intersects the given lines.

### Step 1: Write the equations of the given lines

The first line is given by:
\[ \frac{x-7}{3} = \frac{y-5}{2} = \frac{z-3}{1} \]
Let \( R_1 = t \). Then we can express the coordinates of points on this line as:
\[ x = 3t + 7, \quad y = 2t + 5, \quad z = t + 3 \]

The second line is given by:
\[ \frac{x-1}{2} = \frac{y+1}{4} = \frac{z+1}{3} \]
Let \( R_2 = s \). Then we can express the coordinates of points on this line as:
\[ x = 2s + 1, \quad y = 4s - 1, \quad z = 3s - 1 \]

### Step 2: Set up the equations for intersection

Since both lines must intersect with the line having direction ratios \( 2:2:1 \), we can express the parametric equations of the line with direction ratios \( 2:2:1 \) as:
\[ x = 2k + x_0, \quad y = 2k + y_0, \quad z = k + z_0 \]
where \( (x_0, y_0, z_0) \) is a point on the line.

### Step 3: Equate the coordinates

For the first line:
\[ 3t + 7 = 2k + x_0 \]
\[ 2t + 5 = 2k + y_0 \]
\[ t + 3 = k + z_0 \]

For the second line:
\[ 2s + 1 = 2k + x_0 \]
\[ 4s - 1 = 2k + y_0 \]
\[ 3s - 1 = k + z_0 \]

### Step 4: Solve the equations

From the first line equations, we can express \( k \) in terms of \( t \):
1. From \( t + 3 = k + z_0 \), we get \( k = t + 3 - z_0 \).

Substituting \( k \) into the other equations will allow us to find \( t \) and \( k \).

From the second line equations, we can express \( k \) in terms of \( s \):
1. From \( 3s - 1 = k + z_0 \), we get \( k = 3s - 1 - z_0 \).

Setting both expressions for \( k \) equal gives us a system of equations to solve for \( t \) and \( s \).

### Step 5: Calculate the coordinates of points A and B

After solving the system of equations, we will find the values of \( t \) and \( s \). We can then substitute these values back into the equations for \( A \) and \( B \) to find their coordinates.

### Step 6: Calculate the distance AB

The distance \( AB \) can be calculated using the distance formula:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2} \]