The equation of common tangent to the parabola's `y^(2)=32x and x^(2)=108y` is

Options

A. `2x+3y+12=0`
B. `2x+3y+36=0`
C. `2x+3y-36=0`
D. `2x+3y-12=0`

Answer

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

### Step 1: Identify the parameters of the parabolas
1. The first parabola \( y^2 = 32x \) can be rewritten in the standard form \( y^2 = 4ax \) where \( 4a = 32 \). Thus, \( a = 8 \).
2. The second parabola \( x^2 = 108y \) can be rewritten in the standard form \( x^2 = 4by \) where \( 4b = 108 \). Thus, \( b = 27 \).

### Step 2: Write the equations of the tangents
1. For the first parabola \( y^2 = 32x \), the equation of the tangent line can be expressed as:
\[ y = mx + a \cdot m \]
Substituting \( a = 8 \), we get:
\[ y = mx + 8m \]
This is Equation (1).

2. For the second parabola \( x^2 = 108y \), the equation of the tangent line can be expressed as:
\[ y = mx - b \cdot m^2 \]
Substituting \( b = 27 \), we get:
\[ y = mx - 27m^2 \]
This is Equation (2).

### Step 3: Set the two tangent equations equal to each other
Since both equations represent the same tangent line, we can set them equal to each other:
\[ mx + 8m = mx - 27m^2 \]

### Step 4: Simplify the equation
1. Cancel \( mx \) from both sides:
\[ 8m = -27m^2 \]
2. Rearranging gives:
\[ 27m^2 + 8m = 0 \]
3. Factor out \( m \):
\[ m(27m + 8) = 0 \]
Thus, \( m = 0 \) or \( m = -\frac{8}{27} \).

### Step 5: Find the equation of the tangent line
1. For \( m = -\frac{8}{27} \):
Substitute \( m \) back into either tangent equation. Using Equation (1):
\[ y = -\frac{8}{27}x + 8\left(-\frac{8}{27}\right) \]
Simplifying gives:
\[ y = -\frac{8}{27}x - \frac{64}{27} \]
To eliminate the fraction, multiply through by 27:
\[ 27y = -8x - 64 \]
Rearranging gives:
\[ 8x + 27y + 64 = 0 \]

### Final Answer
The equation of the common tangent to the parabolas \( y^2 = 32x \) and \( x^2 = 108y \) is:
\[ 8x + 27y + 64 = 0 \]