The absolute value of slope of common tangents to parabola `y^(2) = 8x` and hyperbola `3x^(2) -y^(2) =3` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the absolute value of the slope of common tangents to the parabola \( y^2 = 8x \) and the hyperbola \( 3x^2 - y^2 = 3 \), we can follow these steps:

### Step 1: Rewrite the equations in standard form

1. **Hyperbola**: The given equation is \( 3x^2 - y^2 = 3 \).
- Divide by 3 to get:
\[ \frac{x^2}{1} - \frac{y^2}{3} = 1 \]
- This shows that \( a^2 = 1 \) and \( b^2 = 3 \), hence \( a = 1 \) and \( b = \sqrt{3} \).

2. **Parabola**: The given equation is \( y^2 = 8x \).
- This can be rewritten in the form \( y^2 = 4ax \) where \( 4a = 8 \), thus \( a = 2 \).

### Step 2: Write the equations of the tangents

1. **Tangent to the hyperbola**:
- The equation of the tangent to the hyperbola is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
- Substituting \( a = 1 \) and \( b = \sqrt{3} \):
\[ y = mx \pm \sqrt{m^2 - 3} \]

2. **Tangent to the parabola**:
- The equation of the tangent to the parabola is given by:
\[ y = mx + \frac{a}{m} \]
- Substituting \( a = 2 \):
\[ y = mx + \frac{2}{m} \]

### Step 3: Set the equations equal to find common tangents

To find the common tangents, we equate the two tangent equations:
\[ mx \pm \sqrt{m^2 - 3} = mx + \frac{2}{m} \]

### Step 4: Simplify the equation

1. Cancel \( mx \) from both sides:
\[ \pm \sqrt{m^2 - 3} = \frac{2}{m} \]

2. Square both sides to eliminate the square root:
\[ m^2 - 3 = \frac{4}{m^2} \]

3. Multiply through by \( m^2 \) to clear the fraction:
\[ m^4 - 3m^2 - 4 = 0 \]

### Step 5: Let \( t = m^2 \) and solve the quadratic

1. Substitute \( t = m^2 \):
\[ t^2 - 3t - 4 = 0 \]

2. Factor the quadratic:
\[ (t - 4)(t + 1) = 0 \]

3. Solve for \( t \):
\[ t = 4 \quad \text{or} \quad t = -1 \]

Since \( t = m^2 \), we discard \( t = -1 \) (as it gives a complex number) and take \( t = 4 \).

### Step 6: Find the slope \( m \)

1. Since \( t = m^2 = 4 \):
\[ m = \pm 2 \]

### Step 7: Find the absolute value of the slope

The absolute value of the slope of the common tangents is:
\[ |m| = 2 \]

### Final Answer
Thus, the absolute value of the slope of common tangents to the given parabola and hyperbola is \( \boxed{2} \).