To solve the problem, we need to determine the value of \( \frac{b}{4d} \) given that the plane \( ax - by + cz = d \) contains the line defined by the equations \( \frac{x - a}{a} = \frac{y - 2d}{b} = \frac{z - c}{c} \).
### Step-by-Step Solution:
1. **Identify Direction Ratios of the Line:**
The line can be expressed in parametric form:
\[
x = a + at, \quad y = 2d + bt, \quad z = c + ct
\]
The direction ratios of the line are \( (a, b, c) \).
2. **Identify a Point on the Line:**
When \( t = 0 \), the point on the line is:
\[
(x_0, y_0, z_0) = (a, 2d, c)
\]
3. **Substitute the Point into the Plane Equation:**
Since the line lies in the plane, substituting the point into the plane equation \( ax - by + cz = d \) gives:
\[
a(a) - b(2d) + c(c) = d
\]
This simplifies to:
\[
a^2 - 2bd + c^2 = d \quad \text{(Equation 1)}
\]
4. **Use the Direction Ratios:**
For the plane to contain the line, the direction ratios of the line and the normal vector of the plane must be orthogonal. The normal vector of the plane is \( (a, -b, c) \). The condition for orthogonality is:
\[
a \cdot a + (-b) \cdot b + c \cdot c = 0
\]
This simplifies to:
\[
a^2 - b^2 + c^2 = 0 \quad \text{(Equation 2)}
\]
5. **Combine Equations:**
From Equation 2, we can express \( b^2 \) as:
\[
b^2 = a^2 + c^2
\]
Substitute this into Equation 1:
\[
a^2 - 2bd + c^2 = d
\]
Replacing \( b^2 \) gives:
\[
b^2 - 2bd = d
\]
Rearranging this gives:
\[
b^2 - 2bd - d = 0
\]
6. **Solve the Quadratic Equation:**
This is a quadratic in \( b \):
\[
b^2 - 2bd - d = 0
\]
Using the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
Here, \( A = 1, B = -2d, C = -d \):
\[
b = \frac{2d \pm \sqrt{(-2d)^2 - 4 \cdot 1 \cdot (-d)}}{2 \cdot 1}
\]
Simplifying gives:
\[
b = \frac{2d \pm \sqrt{4d^2 + 4d}}{2}
\]
\[
b = d \pm \sqrt{d^2 + d}
\]
7. **Find the Value of \( \frac{b}{4d} \):**
From the earlier derived relationship \( b^2 = 2bd \):
\[
b(b - 2d) = 0
\]
Since \( b \neq 0 \), we have \( b = 2d \). Thus:
\[
\frac{b}{4d} = \frac{2d}{4d} = \frac{1}{2}
\]
### Final Answer:
\[
\frac{b}{4d} = \frac{1}{2}
\]
