The lines which intersect the skew lines `y=m x ,z=c ; y=-m x ,z=-c` and the x-axis lie on the surface
a. `c z=m x y` b. `x y=c m z` c. `c y=m x z` d. none of these

Options

A. `cz=mxy`
B. `xy=cmz`
C. `cy=mxz`
D. none of these

Explanation

To solve the problem, we need to find the surface on which the lines that intersect the given skew lines and the x-axis lie. Let's break down the solution step by step.

### Step 1: Identify the Skew Lines
The given skew lines are:
1. \( y = mx, z = c \)
2. \( y = -mx, z = -c \)

### Step 2: Write the Equations of Planes
We can write the equations of the planes that contain these lines.

For the first line \( y = mx \) and \( z = c \):
- The equation of the plane can be expressed as:
\[ y - mx + \lambda_1 (z - c) = 0 \]
Rearranging gives:
\[ y - mx + \lambda_1 z - \lambda_1 c = 0 \]

For the second line \( y = -mx \) and \( z = -c \):
- The equation of the plane can be expressed as:
\[ y + mx + \lambda_2 (z + c) = 0 \]
Rearranging gives:
\[ y + mx + \lambda_2 z + \lambda_2 c = 0 \]

### Step 3: Find Intersection with the X-axis
To find where these planes intersect the x-axis, we set \( y = 0 \) and \( z = 0 \).

For the first plane:
\[ 0 - mx + \lambda_1 (0 - c) = 0 \implies -mx - \lambda_1 c = 0 \implies \lambda_1 = -\frac{mx}{c} \]

For the second plane:
\[ 0 + mx + \lambda_2 (0 + c) = 0 \implies mx + \lambda_2 c = 0 \implies \lambda_2 = -\frac{mx}{c} \]

### Step 4: Set the Values of \(\lambda_1\) and \(\lambda_2\) Equal
Since both planes intersect at the same point on the x-axis, we have:
\[ \lambda_1 = \lambda_2 \]
This gives us:
\[ -\frac{mx}{c} = -\frac{mx}{c} \quad \text{(which is trivially true)} \]

### Step 5: Derive the Equation of the Surface
From the equations of the planes, we can derive a relationship between \(y\), \(z\), and \(x\). Setting the two equations equal gives:
\[ \frac{y - mx}{z - c} = \frac{y + mx}{z + c} \]

Cross-multiplying yields:
\[ (y - mx)(z + c) = (y + mx)(z - c) \]
Expanding both sides:
\[ yz + cy - mxz - mc = yz - cz + mxz - mc \]
Simplifying leads to:
\[ cy + cz = mxz \]
Rearranging gives:
\[ cy = mxz \]

### Final Equation
Thus, the equation of the surface is:
\[ cy = mxz \]

### Step 6: Check the Options
Comparing with the options provided:
- Option C: \(cy = mxz\) is indeed the correct answer.

### Conclusion
The correct answer is:
**c. \(cy = mxz\)**