The plane containing the line `(x-3)/(2)=(y-b)/(4)=(z-3)/(3)` passes through the points `(a, 1, 2), (2, 1, 4), (2, 3, 5)`, then `3a+5b` is equal to

Options

A. 4
B. 16
C. `-16`
D. `-4`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the plane containing the line

\[ \frac{x-3}{2} = \frac{y-b}{4} = \frac{z-3}{3} \]

passes through the points \( (a, 1, 2) \), \( (2, 1, 4) \), and \( (2, 3, 5) \).

### Step 1: Write the equation of the plane
The line can be represented in parametric form. The direction ratios of the line are \( (2, 4, 3) \). The general equation of the plane can be written as:

\[ 2(x - 3) + 4(y - b) + 3(z - 3) = 0 \]

### Step 2: Expand the plane equation
Expanding the equation gives:

\[ 2x - 6 + 4y - 4b + 3z - 9 = 0 \]

This simplifies to:

\[ 2x + 4y + 3z - 4b - 15 = 0 \]

### Step 3: Substitute the points into the plane equation
We will substitute each of the given points into the plane equation to create a system of equations.

#### For point \( (a, 1, 2) \):
\[ 2a + 4(1) + 3(2) - 4b - 15 = 0 \]
This simplifies to:
\[ 2a + 4 + 6 - 4b - 15 = 0 \implies 2a - 4b - 5 = 0 \quad \text{(Equation 1)} \]

#### For point \( (2, 1, 4) \):
\[ 2(2) + 4(1) + 3(4) - 4b - 15 = 0 \]
This simplifies to:
\[ 4 + 4 + 12 - 4b - 15 = 0 \implies 4 - 4b + 1 = 0 \implies -4b + 1 = 0 \quad \text{(Equation 2)} \]

#### For point \( (2, 3, 5) \):
\[ 2(2) + 4(3) + 3(5) - 4b - 15 = 0 \]
This simplifies to:
\[ 4 + 12 + 15 - 4b - 15 = 0 \implies 16 - 4b = 0 \quad \text{(Equation 3)} \]

### Step 4: Solve the equations
From Equation 2 and Equation 3, we can solve for \( b \):
From Equation 2:
\[ -4b + 1 = 0 \implies 4b = 1 \implies b = \frac{1}{4} \]

From Equation 3:
\[ 16 - 4b = 0 \implies 4b = 16 \implies b = 4 \]

### Step 5: Substitute \( b \) back to find \( a \)
Now we can substitute \( b \) back into Equation 1 to find \( a \):
\[ 2a - 4\left(\frac{1}{4}\right) - 5 = 0 \implies 2a - 1 - 5 = 0 \implies 2a = 6 \implies a = 3 \]

### Step 6: Calculate \( 3a + 5b \)
Now we can calculate \( 3a + 5b \):
\[ 3a + 5b = 3(3) + 5\left(\frac{1}{4}\right) = 9 + \frac{5}{4} = 9 + 1.25 = 10.25 \]

### Final Answer
Thus, the value of \( 3a + 5b \) is \( 10.25 \).