A line passing through P(3, 7, 1) and R(2, 5, 7) meet the plane `3x+2y+11z-9=0` at Q. Then PQ is equal to :

Options

A. `(5sqrt(41))/(59)`
B. `(sqrt(41))/(59)`
C. `(50sqrt(41))/(59)`
D. `(25sqrt(41))/(59)`

Answer

To find the distance \( PQ \) where \( P(3, 7, 1) \) and \( Q \) is the point where the line through points \( P \) and \( R(2, 5, 7) \) intersects the plane given by the equation \( 3x + 2y + 11z - 9 = 0 \), we can follow these steps:

### Step 1: Find the direction ratios of the line PR
The direction ratios of the line passing through points \( P(3, 7, 1) \) and \( R(2, 5, 7) \) can be calculated as follows:

\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (2 - 3, 5 - 7, 7 - 1) = (-1, -2, 6) \]

### Step 2: Write the parametric equations of the line PR
Using the point \( P(3, 7, 1) \) and the direction ratios, we can express the line in parametric form:

\[ x = 3 - t, \quad y = 7 - 2t, \quad z = 1 + 6t \]

### Step 3: Substitute the parametric equations into the plane equation
Now, we substitute \( x, y, z \) into the plane equation \( 3x + 2y + 11z - 9 = 0 \):

\[ 3(3 - t) + 2(7 - 2t) + 11(1 + 6t) - 9 = 0 \]

Expanding this:

\[ 9 - 3t + 14 - 4t + 11 + 66t - 9 = 0 \]

Combining like terms:

\[ (66t - 3t - 4t) + (9 + 14 + 11 - 9) = 0 \]
\[ 59t + 25 = 0 \]

### Step 4: Solve for \( t \)
Now, we solve for \( t \):

\[ 59t = -25 \implies t = -\frac{25}{59} \]

### Step 5: Find the coordinates of point Q
Now we substitute \( t \) back into the parametric equations to find the coordinates of point \( Q \):

\[ x = 3 - \left(-\frac{25}{59}\right) = 3 + \frac{25}{59} = \frac{177 + 25}{59} = \frac{202}{59} \]

\[ y = 7 - 2\left(-\frac{25}{59}\right) = 7 + \frac{50}{59} = \frac{413 + 50}{59} = \frac{463}{59} \]

\[ z = 1 + 6\left(-\frac{25}{59}\right) = 1 - \frac{150}{59} = \frac{59 - 150}{59} = -\frac{91}{59} \]

Thus, the coordinates of point \( Q \) are \( \left(\frac{202}{59}, \frac{463}{59}, -\frac{91}{59}\right) \).

### Step 6: Calculate the distance \( PQ \)
Now, we use the distance formula to find \( PQ \):

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

Substituting the coordinates:

\[ PQ = \sqrt{\left(3 - \frac{202}{59}\right)^2 + \left(7 - \frac{463}{59}\right)^2 + \left(1 + \frac{91}{59}\right)^2} \]

Calculating each term:

1. \( 3 - \frac{202}{59} = \frac{177 - 202}{59} = \frac{-25}{59} \)
2. \( 7 - \frac{463}{59} = \frac{413 - 463}{59} = \frac{-50}{59} \)
3. \( 1 + \frac{91}{59} = \frac{59 + 91}{59} = \frac{150}{59} \)

Thus,

\[ PQ = \sqrt{\left(\frac{-25}{59}\right)^2 + \left(\frac{-50}{59}\right)^2 + \left(\frac{150}{59}\right)^2} \]

Calculating the squares:

\[ PQ = \sqrt{\frac{625}{3481} + \frac{2500}{3481} + \frac{22500}{3481}} = \sqrt{\frac{25625}{3481}} = \frac{25\sqrt{41}}{59} \]

### Final Answer
The distance \( PQ \) is:

\[ PQ = \frac{25\sqrt{41}}{59} \]