If the plane `3x - 4y + 5z = 0` is parallel to `(2x-1)/4=(1-y)/(_3)=(z-2)/a`, then the value of a is

Options

A. `6/4`
B. `6/5`
C. `1.4`
D. `3/4`

Answer

To find the value of \( a \) such that the plane \( 3x - 4y + 5z = 0 \) is parallel to the line given by the equation \( \frac{(2x-1)}{4} = \frac{(1-y)}{(-3)} = \frac{(z-2)}{a} \), we can follow these steps:

### Step 1: Identify the normal vector of the plane
The equation of the plane is given by:
\[ 3x - 4y + 5z = 0 \]
The normal vector \( \mathbf{n} \) to the plane can be extracted from the coefficients of \( x \), \( y \), and \( z \):
\[ \mathbf{n} = (3, -4, 5) \]

### Step 2: Convert the line equation to standard form
The line equation is given in the symmetric form:
\[ \frac{(2x-1)}{4} = \frac{(1-y)}{(-3)} = \frac{(z-2)}{a} \]
We can rewrite this in terms of direction ratios. Let's express it in the form:
\[ \frac{x - \frac{1}{2}}{2} = \frac{y - 1}{-3} = \frac{z - 2}{a} \]
From this, we can identify the direction ratios of the line:
\[ (2, -3, a) \]

### Step 3: Use 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. This means their dot product should be zero:
\[ \mathbf{n} \cdot (2, -3, a) = 0 \]
Calculating the dot product:
\[ 3 \cdot 2 + (-4) \cdot (-3) + 5 \cdot a = 0 \]
This simplifies to:
\[ 6 + 12 + 5a = 0 \]
Combining like terms gives:
\[ 18 + 5a = 0 \]

### Step 4: Solve for \( a \)
Now, we can isolate \( a \):
\[ 5a = -18 \]
\[ a = -\frac{18}{5} \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{-\frac{18}{5}} \]