If a variable line has its intercepts on the coordinate axes `ea n de^(prime),` where `e/2a n d e^(prime)/2` are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle `x^2+y^2=r^2,` where `r=` (a) 1 (b) 2 (c) 3 (d) cannot be decided

Options

A. 1
B. 2
C. 3
D. cannot be decided

Explanation

To solve the problem, we need to analyze the given conditions about the variable line and its relationship with the hyperbola and the circle. Here’s a step-by-step solution:

### Step 1: Understanding the Eccentricities
We are given that the intercepts of a variable line on the coordinate axes are `e` and `e'`, where `e/2` and `e'/2` are the eccentricities of a hyperbola and its conjugate hyperbola.

### Step 2: Using the Eccentricity Condition
The relationship between the eccentricities of a hyperbola and its conjugate hyperbola is given by:
\[ \frac{1}{e^2} + \frac{1}{e'^2} = 1 \]
Substituting `e/2` and `e'/2` into this equation, we have:
\[ \frac{4}{e^2} + \frac{4}{e'^2} = 1 \]
Multiplying through by `e^2 e'^2` gives:
\[ 4e'^2 + 4e^2 = e^2 e'^2 \]

### Step 3: Rearranging the Equation
Rearranging the equation, we have:
\[ e^2 e'^2 - 4e^2 - 4e'^2 = 0 \]

### Step 4: Finding the Equation of the Line
The line with intercepts `e` and `e'` can be expressed in the form:
\[ \frac{x}{e} + \frac{y}{e'} = 1 \]
This can be rewritten as:
\[ e'y + ex = e e' \]

### Step 5: Finding the Distance from the Origin
To find the distance from the origin to the line, we use the formula for the distance from a point to a line:
\[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = e'\), \(B = e\), and \(C = -ee'\). The origin coordinates are \( (0, 0) \):
\[ \text{Distance} = \frac{|0 + 0 - ee'|}{\sqrt{(e')^2 + e^2}} = \frac{ee'}{\sqrt{(e')^2 + e^2}} \]

### Step 6: Setting the Distance Equal to Radius
For the line to always touch the circle \(x^2 + y^2 = r^2\), the distance from the origin to the line must equal the radius \(r\):
\[ \frac{ee'}{\sqrt{(e')^2 + e^2}} = r \]

### Step 7: Finding the Value of r
From the earlier steps, we have established that the distance simplifies to:
\[ r = \frac{2\sqrt{(e')^2 + e^2}}{\sqrt{(e')^2 + e^2}} = 2 \]

### Conclusion
Thus, the radius \(r\) is equal to 2. Therefore, the answer is:
\[ \boxed{2} \]