A symmetrical form of the line of intersection of the planes `x=ay+b` and `z=cy+d` is

Options

A. `(x-b)/a=(y-1)/0=(z-d)/c`
B. `(x-b)/a=(y-0)/1=(z-d)/c`
C. `(x-a)/b=(y-0)/1=(z-c)/d`
D. none of these

Explanation

To find the symmetrical form of the line of intersection of the planes given by the equations \( x = ay + b \) and \( z = cy + d \), we can follow these steps:

### Step 1: Express \( y \) in terms of \( x \)
From the first plane equation \( x = ay + b \), we can rearrange it to express \( y \):
\[ y = \frac{x - b}{a} \]

**Hint:** Rearranging equations to isolate a variable is a common technique in solving systems of equations.

### Step 2: Substitute \( y \) into the second plane equation
Now, we substitute the expression for \( y \) into the second plane equation \( z = cy + d \):
\[ z = c\left(\frac{x - b}{a}\right) + d \]

**Hint:** Substituting known values into equations helps to eliminate variables and find relationships between the remaining variables.

### Step 3: Simplify the equation for \( z \)
Now, simplify the equation:
\[ z = \frac{c(x - b)}{a} + d \]
\[ z = \frac{cx - cb}{a} + d \]
To combine the terms, we can express \( d \) in terms of a common denominator:
\[ z = \frac{cx - cb + ad}{a} \]

**Hint:** When combining fractions, ensure that you have a common denominator to simplify correctly.

### Step 4: Rearranging the equation
Now, rearranging gives us:
\[ az = cx - cb + ad \]
\[ az - cx + cb - ad = 0 \]

**Hint:** Rearranging terms helps to see the relationship between the variables more clearly.

### Step 5: Write in symmetrical form
Now, we can express the equations in a symmetrical form. We have:
\[ \frac{x - b}{a} = \frac{y}{1} = \frac{z - d}{c} \]

**Hint:** Symmetrical forms are useful for visualizing the relationships between multiple variables.

### Final Result
Thus, the symmetrical form of the line of intersection of the given planes is:
\[ \frac{x - b}{a} = \frac{y}{1} = \frac{z - d}{c} \]

### Summary of Steps
1. Isolate \( y \) from the first plane equation.
2. Substitute \( y \) into the second plane equation.
3. Simplify the equation for \( z \).
4. Rearrange the equation to highlight the relationships.
5. Write the final result in symmetrical form.