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, we need to find the sum of the intercepts of a plane on the coordinate axes. The plane passes through the intersection of two given planes and is parallel to a specified line. Here’s a step-by-step solution:

### Step 1: Find the equation of the plane through the intersection of the two given planes.

The two planes are:
1. \( P_1: 2x + 3y + 3z - 5 = 0 \)
2. \( P_2: 2x - 5y + 3z + 1 = 0 \)

The equation of the family of planes through the intersection of these two 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 2: Determine the direction ratios of the line parallel to the plane.

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

### Step 3: Set up the condition for the plane to be parallel to the line.

For the plane to be parallel to the line, the normal vector of the plane must be orthogonal to the direction ratios of the line. The normal vector from the plane equation is:
\[ (2 + 2\lambda, 3 - 5\lambda, 3 + 3\lambda) \]
We need to take the dot product with the direction ratios of the line:
\[ (2 + 2\lambda) \cdot 2 + (3 - 5\lambda)(-5) + (3 + 3\lambda)(-7) = 0 \]
Expanding this gives:
\[ 4 + 4\lambda - 15 + 25\lambda - 21 - 21\lambda = 0 \]
Combining like terms:
\[ (4\lambda + 25\lambda - 21\lambda) + (4 - 15 - 21) = 0 \]
This simplifies to:
\[ 8\lambda - 32 = 0 \]
Thus, we have:
\[ \lambda = 4 \]

### Step 4: Substitute \(\lambda\) back to find 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 results in:
\[ 10x - 17y + 15z - 1 = 0 \]

### Step 5: Find the intercepts on the coordinate axes.

To find the x-intercept, set \(y = 0\) and \(z = 0\):
\[ 10x - 1 = 0 \implies x = \frac{1}{10} \]

To find the y-intercept, set \(x = 0\) and \(z = 0\):
\[ -17y - 1 = 0 \implies y = -\frac{1}{17} \]

To find the z-intercept, set \(x = 0\) and \(y = 0\):
\[ 15z - 1 = 0 \implies z = \frac{1}{15} \]

### Step 6: Calculate the sum of the intercepts.

The sum of the intercepts is:
\[ \frac{1}{10} - \frac{1}{17} + \frac{1}{15} \]

### Step 7: Find a common denominator and simplify.

The least common multiple of \(10, 17, 15\) is \(510\).
Converting each term:
\[ \frac{1}{10} = \frac{51}{510}, \quad -\frac{1}{17} = -\frac{30}{510}, \quad \frac{1}{15} = \frac{34}{510} \]
Adding these:
\[ \frac{51 - 30 + 34}{510} = \frac{55}{510} = \frac{11}{102} \]

Thus, the sum of the intercepts of the plane on the coordinate axes is:
\[ \boxed{\frac{11}{102}} \]