If the equation of the plane passing through the line of intersection of the planes `2x -y+z=3, 4x-3y + 5z + 9 = 0` and parallel to the line `(x+1)(-2) =(y+3)/4 = (z-2)/5` is ax + by + cz + 6 = 0, then a+ b + c is equal to

Options

A. 12
B. 14
C. 15
D. 13

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given planes
The equations of the planes are:
1. \( 2x - y + z = 3 \) (Plane 1)
2. \( 4x - 3y + 5z + 9 = 0 \) (Plane 2)

### Step 2: Write the equation of the plane through the line of intersection
The equation of the plane passing through the line of intersection of the two given planes can be expressed as:
\[ (2x - y + z - 3) + \lambda(4x - 3y + 5z + 9) = 0 \]
where \( \lambda \) is a parameter.

### Step 3: Expand the equation
Expanding the equation gives:
\[ 2x - y + z - 3 + \lambda(4x - 3y + 5z + 9) = 0 \]
\[ (2 + 4\lambda)x + (-1 - 3\lambda)y + (1 + 5\lambda)z + (-3 + 9\lambda) = 0 \]

### Step 4: Identify the coefficients
From the expanded equation, we can identify the coefficients:
- Coefficient of \( x \): \( 2 + 4\lambda \)
- Coefficient of \( y \): \( -1 - 3\lambda \)
- Coefficient of \( z \): \( 1 + 5\lambda \)
- Constant term: \( -3 + 9\lambda \)

### Step 5: Determine the direction ratios of the given line
The line is given in the symmetric form:
\[ \frac{x + 1}{-2} = \frac{y + 3}{4} = \frac{z - 2}{5} \]
The direction ratios of the line are \( (-2, 4, 5) \).

### Step 6: Set up the condition for parallelism
For the plane to be parallel to the line, the normal vector of the plane must be perpendicular to the direction ratios of the line. Thus, we set up the equation:
\[ -2(2 + 4\lambda) + 4(-1 - 3\lambda) + 5(1 + 5\lambda) = 0 \]

### Step 7: Simplify the equation
Expanding and simplifying:
\[ -4 - 8\lambda - 4 + 12\lambda + 5 + 25\lambda = 0 \]
\[ (-4 - 4 + 5) + (-8\lambda + 12\lambda + 25\lambda) = 0 \]
\[ -3 + 29\lambda = 0 \]
Thus, we find:
\[ 29\lambda = 3 \implies \lambda = \frac{3}{29} \]

### Step 8: Substitute \( \lambda \) back into the plane equation
Substituting \( \lambda = \frac{3}{29} \) into the coefficients:
- Coefficient of \( x \): \( 2 + 4\left(\frac{3}{29}\right) = 2 + \frac{12}{29} = \frac{58 + 12}{29} = \frac{70}{29} \)
- Coefficient of \( y \): \( -1 - 3\left(\frac{3}{29}\right) = -1 - \frac{9}{29} = -\frac{29 + 9}{29} = -\frac{38}{29} \)
- Coefficient of \( z \): \( 1 + 5\left(\frac{3}{29}\right) = 1 + \frac{15}{29} = \frac{29 + 15}{29} = \frac{44}{29} \)

### Step 9: Write the final plane equation
The equation of the plane can be written as:
\[ \frac{70}{29}x - \frac{38}{29}y + \frac{44}{29}z + C = 0 \]
To match the form \( ax + by + cz + 6 = 0 \), we can multiply through by 29 to eliminate the fractions:
\[ 70x - 38y + 44z + 29C = 0 \]

### Step 10: Find the constant term
To find \( C \), we can set \( 29C = 6 \) which gives \( C = \frac{6}{29} \).

### Step 11: Identify \( a, b, c \)
From the equation \( 70x - 38y + 44z + 6 = 0 \), we have:
- \( a = 70 \)
- \( b = -38 \)
- \( c = 44 \)

### Step 12: Calculate \( a + b + c \)
Now, we calculate:
\[ a + b + c = 70 - 38 + 44 = 76 \]

### Final Answer
Thus, the value of \( a + b + c \) is \( 76 \).