The distance between the planes `3x+5y+z=8` and `3x+5y+z+27 = 0` is :

Options

A. `(8)/(sqrt(35))`
B. `(27)/(sqrt(35))`
C. `sqrt(35)`
D. `2sqrt(35)`

Answer

To find the distance between the two planes given by the equations \(3x + 5y + z = 8\) and \(3x + 5y + z + 27 = 0\), we can follow these steps:

### Step 1: Rewrite the equations of the planes in standard form
The first plane can be rewritten as:
\[ 3x + 5y + z - 8 = 0 \]
This gives us \(d_1 = -8\).

The second plane can be rewritten as:
\[ 3x + 5y + z + 27 = 0 \]
This gives us \(d_2 = 27\).

### Step 2: Identify the coefficients of the planes
From the equations of the planes, we can identify the coefficients:
- For both planes, the coefficients are \(a_1 = 3\), \(b_1 = 5\), and \(c_1 = 1\).

### Step 3: Check if the planes are parallel
The planes are parallel if their normal vectors are proportional. Since both planes have the same coefficients for \(x\), \(y\), and \(z\), they are indeed parallel.

### Step 4: Use the formula for the distance between two parallel planes
The distance \(D\) between two parallel planes given by the equations \(a_1x + b_1y + c_1z + d_1 = 0\) and \(a_2x + b_2y + c_2z + d_2 = 0\) is given by the formula:
\[ D = \frac{|d_2 - d_1|}{\sqrt{a_1^2 + b_1^2 + c_1^2}} \]

### Step 5: Substitute the values into the formula
Substituting the values we found:
- \(d_1 = -8\)
- \(d_2 = 27\)
- \(a_1 = 3\), \(b_1 = 5\), \(c_1 = 1\)

We get:
\[ D = \frac{|27 - (-8)|}{\sqrt{3^2 + 5^2 + 1^2}} = \frac{|27 + 8|}{\sqrt{9 + 25 + 1}} = \frac{35}{\sqrt{35}} \]

### Step 6: Simplify the expression
We can simplify \(D\):
\[ D = \frac{35}{\sqrt{35}} = \sqrt{35} \]

### Conclusion
Thus, the distance between the two planes is:
\[ \sqrt{35} \]

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