The equation of plane passing through the intersection of the planes `2x+y+2z=9,4x-5y-4z=1` and the point `(3,2,1)` is

Options

A. `10x-2y+2z=28`
B. `10x+2y+2z=28`
C. `10x-2y-2z=28`
D. `10x-2y-2z=24`

Answer

To find the equation of the plane passing through the intersection of the planes \(2x + y + 2z = 9\) and \(4x - 5y - 4z = 1\) and the point \((3, 2, 1)\), we can follow these steps:

### Step 1: Write the equations of the given planes
We have the equations of two planes:
1. Plane 1: \(2x + y + 2z = 9\)
2. Plane 2: \(4x - 5y - 4z = 1\)

### Step 2: Form the equation of the required plane
The equation of the plane passing through the intersection of the two given planes can be expressed as:
\[ 2x + y + 2z - 9 + \lambda(4x - 5y - 4z - 1) = 0 \]
where \(\lambda\) is a parameter.

### Step 3: Substitute the point \((3, 2, 1)\) into the equation
We need to ensure that the point \((3, 2, 1)\) satisfies the equation of the plane. Substituting \(x = 3\), \(y = 2\), and \(z = 1\) into the equation:
\[ 2(3) + 2 + 2(1) - 9 + \lambda(4(3) - 5(2) - 4(1) - 1) = 0 \]

### Step 4: Simplify the equation
Calculating the left-hand side:
\[ 6 + 2 + 2 - 9 + \lambda(12 - 10 - 4 - 1) = 0 \]
This simplifies to:
\[ 1 + \lambda(-3) = 0 \]
Thus, we have:
\[ 1 - 3\lambda = 0 \]

### Step 5: Solve for \(\lambda\)
From the equation \(1 - 3\lambda = 0\), we can solve for \(\lambda\):
\[ 3\lambda = 1 \implies \lambda = \frac{1}{3} \]

### Step 6: Substitute \(\lambda\) back into the plane equation
Now, substitute \(\lambda = \frac{1}{3}\) back into the equation of the plane:
\[ 2x + y + 2z - 9 + \frac{1}{3}(4x - 5y - 4z - 1) = 0 \]

### Step 7: Clear the fraction by multiplying through by 3
Multiplying the entire equation by 3 to eliminate the fraction:
\[ 3(2x + y + 2z - 9) + (4x - 5y - 4z - 1) = 0 \]
This expands to:
\[ 6x + 3y + 6z - 27 + 4x - 5y - 4z - 1 = 0 \]

### Step 8: Combine like terms
Combining the like terms gives:
\[ (6x + 4x) + (3y - 5y) + (6z - 4z) - 28 = 0 \]
This simplifies to:
\[ 10x - 2y + 2z - 28 = 0 \]

### Step 9: Write the final equation
Rearranging gives us the final equation of the plane:
\[ 10x - 2y + 2z = 28 \]

### Final Answer
The equation of the plane is:
\[ 10x - 2y + 2z = 28 \]