The planes: `2x y + 4z = 5 a n d 5x 2. 5 y + 10 z = 6`are(A) Perpendicular (B) Parallel(C) intersect y-axis (D) passes through `(0,0,5/4)`

Options

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

Answer

To determine the relationship between the two given planes, we will analyze their equations and properties step by step.

### Step 1: Write down the equations of the planes
The equations of the planes are given as:
1. Plane 1: \( 2x - y + 4z = 5 \)
2. Plane 2: \( 5x - 2.5y + 10z = 6 \)

### Step 2: Identify the normal vectors of the planes
The normal vector of a plane given by the equation \( Ax + By + Cz = D \) is \( \vec{n} = (A, B, C) \).

For Plane 1:
- Coefficients are \( A = 2, B = -1, C = 4 \)
- Normal vector \( \vec{n_1} = (2, -1, 4) \)

For Plane 2:
- Coefficients are \( A = 5, B = -2.5, C = 10 \)
- Normal vector \( \vec{n_2} = (5, -2.5, 10) \)

### Step 3: Check if the normal vectors are parallel
Two vectors are parallel if one is a scalar multiple of the other.

Let's express \( \vec{n_2} \) in terms of \( \vec{n_1} \):
- \( \vec{n_2} = (5, -2.5, 10) \)
- We can factor out \( 2.5 \) from \( \vec{n_2} \):
\[ \vec{n_2} = 2.5 \cdot (2, -1, 4) = 2.5 \cdot \vec{n_1} \]

Since \( \vec{n_2} \) is a scalar multiple of \( \vec{n_1} \), the planes are parallel.

### Step 4: Check if the planes intersect or are parallel
Since the planes are parallel, they do not intersect unless they are the same plane. To check if they are the same plane, we can compare the ratios of the coefficients.

For Plane 1:
- Coefficients: \( 2, -1, 4 \) and constant \( 5 \)

For Plane 2:
- Coefficients: \( 5, -2.5, 10 \) and constant \( 6 \)

The ratios of the coefficients of the planes are:
\[ \frac{2}{5} \neq \frac{-1}{-2.5} \neq \frac{4}{10} \neq \frac{5}{6} \]
Since the ratios are not equal, the planes are not the same and therefore do not intersect.

### Step 5: Check if the planes pass through the point \( (0, 0, \frac{5}{4}) \)
We need to substitute the point \( (0, 0, \frac{5}{4}) \) into both plane equations.

For Plane 1:
\[ 2(0) - (0) + 4\left(\frac{5}{4}\right) = 5 \implies 0 + 0 + 5 = 5 \quad \text{(True)} \]

For Plane 2:
\[ 5(0) - 2.5(0) + 10\left(\frac{5}{4}\right) = 6 \implies 0 + 0 + 12.5 \neq 6 \quad \text{(False)} \]

### Conclusion
- The planes are parallel.
- They do not intersect.
- The point \( (0, 0, \frac{5}{4}) \) lies on Plane 1 but not on Plane 2.

Thus, the correct answer is that the planes are **(B) Parallel**.