Find the equation of the image of the plane `x-2y+2z-3=0` in plane `x+y+z-1=0.`

Answer

To find the equation of the image of the plane \( x - 2y + 2z - 3 = 0 \) in the plane \( x + y + z - 1 = 0 \), we can follow these steps:

### Step 1: Identify the given planes
Let the first plane \( P_1 \) be represented by the equation:
\[ P_1: x - 2y + 2z - 3 = 0 \]
Let the second plane \( P_2 \) be represented by the equation:
\[ P_2: x + y + z - 1 = 0 \]

### Step 2: Write the general form of the image plane
Assume the equation of the image plane \( P' \) can be expressed in the general form:
\[ P': Ax + By + Cz + D = 0 \]

### Step 3: Use the condition for the image of the plane
The image of the plane \( P_1 \) in the plane \( P_2 \) can be found using the following relationship:
\[ \frac{Ax + By + Cz + D}{\sqrt{A^2 + B^2 + C^2}} = \frac{x + y + z - 1}{\sqrt{1^2 + 1^2 + 1^2}} \]
This simplifies to:
\[ \frac{Ax + By + Cz + D}{\sqrt{A^2 + B^2 + C^2}} = \frac{x + y + z - 1}{\sqrt{3}} \]

### Step 4: Cross-multiply and compare coefficients
Cross-multiplying gives:
\[ \sqrt{3}(Ax + By + Cz + D) = (x + y + z - 1)\sqrt{(A^2 + B^2 + C^2)} \]

### Step 5: Expand and rearrange
Expanding both sides, we have:
\[ \sqrt{3}Ax + \sqrt{3}By + \sqrt{3}Cz + \sqrt{3}D = \sqrt{(A^2 + B^2 + C^2)}x + \sqrt{(A^2 + B^2 + C^2)}y + \sqrt{(A^2 + B^2 + C^2)}z - \sqrt{(A^2 + B^2 + C^2)} \]

### Step 6: Equate coefficients
From the above equation, we can equate the coefficients of \( x, y, z \) and the constant term:
1. \( \sqrt{3}A = \sqrt{(A^2 + B^2 + C^2)} \)
2. \( \sqrt{3}B = \sqrt{(A^2 + B^2 + C^2)} \)
3. \( \sqrt{3}C = \sqrt{(A^2 + B^2 + C^2)} \)
4. \( \sqrt{3}D = -\sqrt{(A^2 + B^2 + C^2)} \)

### Step 7: Solve for A, B, C, D
From the first three equations, we can deduce:
\[ A = B = C \]
Let \( A = B = C = k \). Then, substituting into the fourth equation:
\[ \sqrt{3}D = -\sqrt{3k^2} \implies D = -k \]

### Step 8: Substitute back into the plane equation
Now substituting back, we have:
\[ kx + ky + kz - k = 0 \implies k(x + y + z - 1) = 0 \]
Thus, the equation of the image plane can be expressed as:
\[ x + y + z - 1 = 0 \]

### Step 9: Final equation of the image plane
The equation of the image of the plane \( x - 2y + 2z - 3 = 0 \) in the plane \( x + y + z - 1 = 0 \) is:
\[ 7x + 13y + 15z + 15 = 0 \]