Distance between two parallel planes `2x""+""y""+""2z""=""8` and `4x""+""2y""+""4z""+""5""=""0` is (1) `5/2` (2) `7/2` (3) `9/2` (4) `3/2`

Options

A. `9//2`
B. `5//2`
C. `7//2`
D. `3//2`

Explanation

To find the distance between the two parallel planes given by the equations \(2x + y + 2z = 8\) and \(4x + 2y + 4z + 5 = 0\), we can follow these steps:

### Step 1: Rewrite the equations in standard form
The first plane is already in the standard form:
\[ 2x + y + 2z - 8 = 0 \]
For the second plane, we can rewrite it as:
\[ 4x + 2y + 4z + 5 = 0 \]
To make the coefficients of \(x\), \(y\), and \(z\) the same, we can divide the entire equation by 2:
\[ 2x + y + 2z + \frac{5}{2} = 0 \]

### Step 2: Identify coefficients and constants
Now, we can identify the coefficients and the constants for both planes:
- For the first plane \(2x + y + 2z - 8 = 0\):
- \(a = 2\), \(b = 1\), \(c = 2\), \(d_1 = -8\)

- For the second plane \(2x + y + 2z + \frac{5}{2} = 0\):
- \(a = 2\), \(b = 1\), \(c = 2\), \(d_2 = \frac{5}{2}\)

### Step 3: Use the distance formula
The formula for the distance \(D\) between two parallel planes \(ax + by + cz + d_1 = 0\) and \(ax + by + cz + d_2 = 0\) is given by:
\[ D = \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ D = \frac{|-8 - \frac{5}{2}|}{\sqrt{2^2 + 1^2 + 2^2}} \]

### Step 5: Simplify the numerator
First, calculate \(d_1 - d_2\):
\[ -8 - \frac{5}{2} = -\frac{16}{2} - \frac{5}{2} = -\frac{21}{2} \]
Taking the absolute value:
\[ |-8 - \frac{5}{2}| = \frac{21}{2} \]

### Step 6: Simplify the denominator
Now, calculate the denominator:
\[ \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Step 7: Final calculation
Now, substituting back into the distance formula:
\[ D = \frac{\frac{21}{2}}{3} = \frac{21}{6} = \frac{7}{2} \]

### Conclusion
The distance between the two parallel planes is \(\frac{7}{2}\).