The planes : 2x - y + 4z = 5 and 5x - 2.5y + 10z = 6 are

Options

A. Perpendicualr
B. Parallel
C. Intersect y-axis
D. Passes through `(0,0(5)/(4))`

Answer

To determine the relationship between the two planes given by the equations \(2x - y + 4z = 5\) and \(5x - 2.5y + 10z = 6\), we will follow these steps:

### Step 1: Identify the coefficients of the planes
The general form of a plane is given by \(Ax + By + Cz + D = 0\). We will rewrite both equations in this form and identify the coefficients.

1. **First Plane:**
\[ 2x - y + 4z - 5 = 0 \]
Here, \(A_1 = 2\), \(B_1 = -1\), \(C_1 = 4\), and \(D_1 = -5\).

2. **Second Plane:**
\[ 5x - 2.5y + 10z - 6 = 0 \]
Here, \(A_2 = 5\), \(B_2 = -2.5\), \(C_2 = 10\), and \(D_2 = -6\).

### Step 2: Calculate the ratios of the coefficients
To determine the relationship between the two planes, we will calculate the ratios of the coefficients.

- Calculate \( \frac{A_1}{A_2} \):
\[ \frac{A_1}{A_2} = \frac{2}{5} \]

- Calculate \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{-1}{-2.5} = \frac{10}{25} = \frac{2}{5} \]

- Calculate \( \frac{C_1}{C_2} \):
\[ \frac{C_1}{C_2} = \frac{4}{10} = \frac{2}{5} \]

### Step 3: Analyze the ratios
Now we compare the ratios we calculated:

\[ \frac{A_1}{A_2} = \frac{2}{5}, \quad \frac{B_1}{B_2} = \frac{2}{5}, \quad \frac{C_1}{C_2} = \frac{2}{5} \]

Since all three ratios are equal, we conclude that the planes are parallel.

### Conclusion
The two planes \(2x - y + 4z = 5\) and \(5x - 2.5y + 10z = 6\) are parallel to each other.

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