If `y=mx+4` is a tangent to both the parabolas, `y^(2)=4x` and `x^(2)=2by`, then b is equal to :

Options

A. `-64`
B. 128
C. `-128`
D. `-32`

Answer

To solve the problem, we need to determine the value of \( b \) such that the line \( y = mx + 4 \) is a tangent to both parabolas \( y^2 = 4x \) and \( x^2 = 2by \).

### Step 1: Tangent to the first parabola \( y^2 = 4x \)

The equation of the parabola \( y^2 = 4x \) can be expressed in the standard form, where \( a = 1 \). The equation of the tangent line to this parabola at a point can be given by:

\[ y = mx + \frac{1}{m} \]

Here, \( \frac{1}{m} \) is the y-intercept of the tangent line.

Since the line \( y = mx + 4 \) is a tangent to this parabola, we can equate the y-intercepts:

\[ \frac{1}{m} = 4 \]

From this, we can solve for \( m \):

\[ m = \frac{1}{4} \]

### Step 2: Tangent to the second parabola \( x^2 = 2by \)

The second parabola \( x^2 = 2by \) can be rewritten in the standard form \( y = \frac{x^2}{2b} \). The equation of the tangent line to this parabola at a point can be given by:

\[ y = mx - \frac{m^2}{2b} \]

Again, since the line \( y = mx + 4 \) is a tangent to this parabola, we can equate the y-intercepts:

\[ -\frac{m^2}{2b} = 4 \]

### Step 3: Substitute \( m \) into the equation

Now substituting \( m = \frac{1}{4} \) into the equation:

\[ -\frac{\left(\frac{1}{4}\right)^2}{2b} = 4 \]

This simplifies to:

\[ -\frac{\frac{1}{16}}{2b} = 4 \]

Multiplying both sides by \( -2b \):

\[ \frac{1}{16} = -8b \]

### Step 4: Solve for \( b \)

Now, we can solve for \( b \):

\[ b = -\frac{1}{128} \]

### Final Result

Thus, the value of \( b \) is:

\[ \boxed{-\frac{1}{128}} \]