If the image of the point `P(1,-2,3)` in the plane, `2x+3y-4z+22=0` measured parallel to the line, `x/1=y/4=z/5` is `Q` , then `P Q` is equal to : `sqrt(42)` (2) `6sqrt(5)` (3) `3sqrt(5)` (4) `3sqrt(42)`

Options

A. `3sqrt(5)`
B. `2sqrt(42)`
C. `sqrt(42)`
D. `6sqrt(5)`

Answer

To find the length of \( PQ \) where \( P(1, -2, 3) \) is a point and \( Q \) is its image in the plane \( 2x + 3y - 4z + 22 = 0 \) measured parallel to the line \( \frac{x}{1} = \frac{y}{4} = \frac{z}{5} \), we can follow these steps:

### Step 1: Write the parametric equations of the line
The line is given in the symmetric form \( \frac{x}{1} = \frac{y}{4} = \frac{z}{5} \). We can express this in parametric form:
- Let \( \lambda \) be the parameter.
- Then, the parametric equations are:
\[ x = \lambda + 1, \quad y = 4\lambda - 2, \quad z = 5\lambda + 3 \]

### Step 2: Substitute into the plane equation
Since point \( R(\lambda + 1, 4\lambda - 2, 5\lambda + 3) \) lies on the plane, we substitute these coordinates into the plane equation \( 2x + 3y - 4z + 22 = 0 \):
\[ 2(\lambda + 1) + 3(4\lambda - 2) - 4(5\lambda + 3) + 22 = 0 \]
Expanding this:
\[ 2\lambda + 2 + 12\lambda - 6 - 20\lambda - 12 + 22 = 0 \]
Combining like terms:
\[ (2 + 12 - 20)\lambda + (2 - 6 - 12 + 22) = 0 \]
This simplifies to:
\[ -6\lambda + 6 = 0 \]
Thus, solving for \( \lambda \):
\[ \lambda = 1 \]

### Step 3: Find the coordinates of point \( R \)
Substituting \( \lambda = 1 \) back into the parametric equations:
\[ x = 1 + 1 = 2, \quad y = 4(1) - 2 = 2, \quad z = 5(1) + 3 = 8 \]
So, the coordinates of point \( R \) are \( (2, 2, 8) \).

### Step 4: Calculate the distance \( PR \)
Using the distance formula between points \( P(1, -2, 3) \) and \( R(2, 2, 8) \):
\[ PR = \sqrt{(2 - 1)^2 + (2 - (-2))^2 + (8 - 3)^2} \]
Calculating each term:
\[ = \sqrt{(1)^2 + (4)^2 + (5)^2} = \sqrt{1 + 16 + 25} = \sqrt{42} \]

### Step 5: Find the distance \( PQ \)
Since \( R \) is the midpoint of \( P \) and \( Q \), we have:
\[ PQ = 2 \cdot PR = 2 \cdot \sqrt{42} = 2\sqrt{42} \]

### Final Answer
Thus, the length of \( PQ \) is \( 2\sqrt{42} \).