If the image of the point `P(1, -2,3)` in the plane `2x+ 3y + 4z + 22 = 0` measured parallel to the line `20x = 5y = 4z` is point Q, then the value of `|PQ|^(2)` is

Answer

To solve the problem, we need to find the square of the distance between the point \( P(1, -2, 3) \) and its image \( Q \) in the plane given by the equation \( 2x + 3y + 4z + 22 = 0 \), where the image is measured parallel to the line defined by \( 20x = 5y = 4z \).

### Step-by-step Solution:

1. **Identify the Direction Ratios of the Line**:
The line \( 20x = 5y = 4z \) can be expressed in parametric form:
\[ x = \frac{1}{20}t, \quad y = \frac{1}{5}t, \quad z = \frac{1}{4}t \]
The direction ratios of the line are \( (1, 4, 5) \).

2. **Equation of the Line through Point P**:
The line through point \( P(1, -2, 3) \) in the direction of the line's direction ratios can be written as:
\[ \frac{x - 1}{1} = \frac{y + 2}{4} = \frac{z - 3}{5} = t \]
From this, we can express \( x, y, z \) in terms of \( t \):
\[ x = t + 1, \quad y = 4t - 2, \quad z = 5t + 3 \]

3. **Substituting into the Plane Equation**:
Substitute \( x, y, z \) into the plane equation \( 2x + 3y + 4z + 22 = 0 \):
\[ 2(t + 1) + 3(4t - 2) + 4(5t + 3) + 22 = 0 \]
Simplifying this:
\[ 2t + 2 + 12t - 6 + 20t + 12 + 22 = 0 \]
Combine like terms:
\[ (2t + 12t + 20t) + (2 - 6 + 12 + 22) = 0 \]
\[ 34t + 30 = 0 \]
Thus, solving for \( t \):
\[ 34t = -30 \implies t = -\frac{30}{34} = -\frac{15}{17} \]

4. **Finding the Coordinates of Point Q**:
Substitute \( t = -\frac{15}{17} \) back into the equations for \( x, y, z \):
\[ x = -\frac{15}{17} + 1 = \frac{2}{17}, \quad y = 4\left(-\frac{15}{17}\right) - 2 = -\frac{60}{17} - \frac{34}{17} = -\frac{94}{17}, \quad z = 5\left(-\frac{15}{17}\right) + 3 = -\frac{75}{17} + \frac{51}{17} = -\frac{24}{17} \]
Therefore, the coordinates of point \( Q \) are:
\[ Q\left(\frac{2}{17}, -\frac{94}{17}, -\frac{24}{17}\right) \]

5. **Calculating the Distance \( |PQ|^2 \)**:
Now we calculate the square of the distance \( |PQ|^2 \):
\[ |PQ|^2 = (x_Q - x_P)^2 + (y_Q - y_P)^2 + (z_Q - z_P)^2 \]
Substituting the coordinates:
\[ |PQ|^2 = \left(\frac{2}{17} - 1\right)^2 + \left(-\frac{94}{17} + 2\right)^2 + \left(-\frac{24}{17} - 3\right)^2 \]
Simplifying each term:
\[ |PQ|^2 = \left(\frac{2 - 17}{17}\right)^2 + \left(-\frac{94 - 34}{17}\right)^2 + \left(-\frac{24 - 51}{17}\right)^2 \]
\[ = \left(-\frac{15}{17}\right)^2 + \left(-\frac{60}{17}\right)^2 + \left(-\frac{75}{17}\right)^2 \]
\[ = \frac{225}{289} + \frac{3600}{289} + \frac{5625}{289} \]
\[ = \frac{225 + 3600 + 5625}{289} = \frac{9450}{289} \]

Thus, the value of \( |PQ|^2 \) is:
\[ \boxed{\frac{9450}{289}} \]