If the parabolas `y^2 = 4b (x - c) " and " y^2 = 8ax` have a common tangent, then which one of the following is a valid choice for the ordered triad (a, b, c)?

Options

A. (1, 1, 1)
B. (1, 3, 2)
C. (1, 3, -2 )
D. (3, 2, 1)

Explanation

To find the ordered triad (a, b, c) for the given parabolas that have a common tangent, we will follow these steps:

### Step 1: Write the equations of the parabolas
The equations of the two parabolas are given as:
1. \( y^2 = 4b(x - c) \)
2. \( y^2 = 8ax \)

### Step 2: Write the equations of the tangents
For the first parabola \( y^2 = 4b(x - c) \), the equation of the tangent in slope form is:
\[ y = mx + \left( \frac{b}{m} - mc \right) \]

For the second parabola \( y^2 = 8ax \), the equation of the tangent in slope form is:
\[ y = mx + \frac{2a}{m} \]

### Step 3: Set the tangent equations equal
Since both parabolas have a common tangent, we can equate the constant terms from both tangent equations:
\[ \frac{b}{m} - mc = \frac{2a}{m} \]

### Step 4: Rearranging the equation
Rearranging the equation gives us:
\[ b - 2a = mc^2 \]

### Step 5: Analyze the conditions for common tangents
For the above equation to hold true, we need to analyze the conditions:
1. \( b - 2a > 0 \) and \( c > 0 \) (Case 1)
2. \( b - 2a < 0 \) and \( c < 0 \) (Case 2)

### Step 6: Check the options
Now we will check the given options for the ordered triad (a, b, c):

1. **Option 1: (1, 1, 1)**
- \( b - 2a = 1 - 2(1) = -1 \) (Case 2, but c is not < 0)

2. **Option 2: (1, 3, 2)**
- \( b - 2a = 3 - 2(1) = 1 \) (Case 1, c > 0)
- This option satisfies the conditions.

3. **Option 3: (1, 3, -2)**
- \( b - 2a = 3 - 2(1) = 1 \) (Case 1, but c is not > 0)

4. **Option 4: (3, 2, 1)**
- \( b - 2a = 2 - 2(3) = -4 \) (Case 2, but c is not < 0)

### Conclusion
The only valid choice for the ordered triad (a, b, c) that satisfies the conditions for having a common tangent is:
**(1, 3, 2)**