The x - intercept of the common tangent to the parabolas `y^(2)=32x and x^(2)=108y` is

Options

A. `-18`
B. `-12`
C. `-9`
D. `-6`

Answer

To find the x-intercept of the common tangent to the parabolas \( y^2 = 32x \) and \( x^2 = 108y \), we will follow these steps:

### Step 1: Write the equations of the parabolas
The first parabola is given by:
\[ y^2 = 32x \quad \text{(C1)} \]
The second parabola is given by:
\[ x^2 = 108y \quad \text{(C2)} \]

### Step 2: Find the equation of the tangent line to the first parabola
The equation of the tangent line to the parabola \( y^2 = 4ax \) is given by:
\[ y = mx + \frac{a}{m} \]
For the parabola \( y^2 = 32x \), we have \( 4a = 32 \) which implies \( a = 8 \). Therefore, the equation of the tangent line is:
\[ y = mx + \frac{8}{m} \]

### Step 3: Substitute the tangent line equation into the second parabola
Now, we substitute \( y \) in the second parabola's equation \( x^2 = 108y \):
\[ x^2 = 108\left(mx + \frac{8}{m}\right) \]
This simplifies to:
\[ x^2 = 108mx + \frac{864}{m} \]
Rearranging gives us:
\[ x^2 - 108mx - \frac{864}{m} = 0 \]

### Step 4: Set the discriminant to zero for a common tangent
For the tangent to be common to both parabolas, the quadratic equation must have exactly one solution. This occurs when the discriminant is zero:
\[ D = b^2 - 4ac = 0 \]
Here, \( a = 1 \), \( b = -108m \), and \( c = -\frac{864}{m} \). Thus, we set up the discriminant:
\[ (-108m)^2 - 4(1)\left(-\frac{864}{m}\right) = 0 \]
Calculating this gives:
\[ 11664m^2 + \frac{3456}{m} = 0 \]
Multiplying through by \( m \) (assuming \( m \neq 0 \)):
\[ 11664m^3 + 3456 = 0 \]
Thus,
\[ m^3 = -\frac{3456}{11664} \]
Simplifying gives:
\[ m^3 = -\frac{8}{27} \]
Taking the cube root:
\[ m = -\frac{2}{3} \]

### Step 5: Substitute the value of \( m \) back into the tangent equation
Now substitute \( m = -\frac{2}{3} \) back into the tangent equation:
\[ y = -\frac{2}{3}x + \frac{8}{-\frac{2}{3}} = -\frac{2}{3}x - 12 \]

### Step 6: Find the x-intercept
To find the x-intercept, set \( y = 0 \):
\[ 0 = -\frac{2}{3}x - 12 \]
Solving for \( x \):
\[ \frac{2}{3}x = -12 \implies x = -12 \cdot \frac{3}{2} = -18 \]

### Final Answer
The x-intercept of the common tangent is:
\[ \boxed{-18} \]