The sum of the intercepts of the plane on the coordinate axes, passing through the intersection of the planes `2x+3y+3z-5=0` and `2x-5y+3z+1=0` and parallel to the line `(x-1)/(2)=(y-2)/(-5)=(z-3)/(-7)`, is

Options

A. `(2)/(5)`
B. `(11)/(105)`
C. `(11)/(102)`
D. `(3)/(101)`

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript and elaborate on each step.

### Step 1: Identify the equations of the planes
We have two planes given by:
1. \( P_1: 2x + 3y + 3z - 5 = 0 \)
2. \( P_2: 2x - 5y + 3z + 1 = 0 \)

### Step 2: Write the equation of the plane through the intersection of the two planes
The equation of a plane passing through the intersection of the two given planes can be expressed as:
\[ P: P_1 + \lambda P_2 = 0 \]
Substituting the equations of the planes:
\[ (2x + 3y + 3z - 5) + \lambda(2x - 5y + 3z + 1) = 0 \]
This simplifies to:
\[ (2 + 2\lambda)x + (3 - 5\lambda)y + (3 + 3\lambda)z + (-5 + \lambda) = 0 \]

### Step 3: Identify the direction ratios of the line
The line is given by:
\[ \frac{x - 1}{2} = \frac{y - 2}{-5} = \frac{z - 3}{-7} \]
The direction ratios of this line are \( (2, -5, -7) \).

### Step 4: 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. The normal vector of the plane is given by:
\[ (2 + 2\lambda, 3 - 5\lambda, 3 + 3\lambda) \]
We apply the condition for perpendicularity:
\[ (2 + 2\lambda) \cdot 2 + (3 - 5\lambda)(-5) + (3 + 3\lambda)(-7) = 0 \]

### Step 5: Expand and simplify the equation
Expanding the equation:
\[ 4 + 4\lambda - 15 + 25\lambda - 21 - 21\lambda = 0 \]
Combining like terms:
\[ (4\lambda + 25\lambda - 21\lambda) + (4 - 15 - 21) = 0 \]
This simplifies to:
\[ 8\lambda - 32 = 0 \]

### Step 6: Solve for \(\lambda\)
Solving for \(\lambda\):
\[ 8\lambda = 32 \implies \lambda = 4 \]

### Step 7: Substitute \(\lambda\) back into the plane equation
Substituting \(\lambda = 4\) into the plane equation:
\[ (2 + 2 \cdot 4)x + (3 - 5 \cdot 4)y + (3 + 3 \cdot 4)z + (-5 + 4) = 0 \]
This gives:
\[ 10x - 17y + 15z - 1 = 0 \]

### Step 8: Write the intercept form of the plane
The intercept form of the plane is:
\[ \frac{x}{\frac{1}{10}} + \frac{y}{-\frac{1}{17}} + \frac{z}{\frac{1}{15}} = 1 \]

### Step 9: Calculate the intercepts
The intercepts on the axes are:
- \( x\)-intercept: \( \frac{1}{10} \)
- \( y\)-intercept: \( -\frac{1}{17} \)
- \( z\)-intercept: \( \frac{1}{15} \)

### Step 10: Find the sum of the intercepts
The sum of the intercepts is:
\[ \frac{1}{10} - \frac{1}{17} + \frac{1}{15} \]
To compute this, we find a common denominator (which is 510):
\[ \frac{51}{510} - \frac{30}{510} + \frac{34}{510} = \frac{55}{510} = \frac{11}{102} \]

### Final Answer
The sum of the intercepts is:
\[ \frac{11}{102} \]