If y=mx+6 is a tangent to both the parabolas `y^2=8x` and `x^2=3by` , then b is equal to :

Options

A. 36
B. `-36`
C. `72`
D. `-72`

Explanation

To solve the problem, we need to find the value of \( b \) such that the line \( y = mx + 6 \) is a tangent to both parabolas \( y^2 = 8x \) and \( x^2 = 3by \).

### Step 1: Identify the parameters of the first parabola
The equation of the first parabola is \( y^2 = 8x \). We can rewrite this in the standard form \( y^2 = 4ax \) where \( a = 2 \) (since \( 4a = 8 \)).

### Step 2: Use the tangent formula for the first parabola
For the parabola \( y^2 = 4ax \), the equation of the tangent line can be expressed as:
\[ y = mx + \frac{a}{m} \]
Substituting \( a = 2 \):
\[ y = mx + \frac{2}{m} \]
Since we know the tangent line is also given by \( y = mx + 6 \), we can equate the two expressions:
\[ \frac{2}{m} = 6 \]

### Step 3: Solve for \( m \)
From the equation \( \frac{2}{m} = 6 \), we can solve for \( m \):
\[ 2 = 6m \implies m = \frac{2}{6} = \frac{1}{3} \]

### Step 4: Write the equation of the tangent line
Now that we have \( m \), we can write the equation of the tangent line:
\[ y = \frac{1}{3}x + 6 \]

### Step 5: Substitute into the second parabola
Next, we need to check if this line is also a tangent to the second parabola \( x^2 = 3by \). We can express \( y \) in terms of \( x \):
\[ y = \frac{1}{3}x + 6 \]
Substituting this into the parabola's equation:
\[ x^2 = 3b\left(\frac{1}{3}x + 6\right) \]
This simplifies to:
\[ x^2 = bx + 18b \]

### Step 6: Rearrange the equation
Rearranging gives us:
\[ x^2 - bx - 18b = 0 \]

### Step 7: Use the discriminant condition for tangency
For the line to be a tangent to the parabola, the discriminant of this quadratic equation must be zero:
\[ D = b^2 - 4ac = 0 \]
Here, \( a = 1 \), \( b = -b \), and \( c = -18b \):
\[ D = (-b)^2 - 4(1)(-18b) = b^2 + 72b \]
Setting the discriminant to zero:
\[ b^2 + 72b = 0 \]

### Step 8: Factor the equation
Factoring gives:
\[ b(b + 72) = 0 \]
This gives us two solutions:
1. \( b = 0 \)
2. \( b = -72 \)

### Step 9: Conclusion
Since we are looking for the value of \( b \) that satisfies the conditions of the problem, we take:
\[ b = -72 \]

### Final Answer
Thus, the value of \( b \) is \( \boxed{-72} \).