The equations of common tangent to the parabola `y^2 = 4ax` and `x^2 = 4by` is

Options

A. `xa^(1//3) + yb^(1//3) + (ab)^(2//3) = 0`
B. `x/(a^(1//3))+y/(b^(1//3))+1/((ab)^(2//3))=0`
C. `xb^(1//3) + ya^(1//3) - (ab)^(2//3) = 0`
D. `x/(b^(1//3))+y/(a^(1//3))+1/((ab)^(2//3))=0`

Answer

To find the equations of the common tangents to the parabolas \( y^2 = 4ax \) and \( x^2 = 4by \), we can follow these steps:

### Step 1: Identify the equations of the parabolas
The given parabolas are:
1. \( y^2 = 4ax \) (which opens to the right)
2. \( x^2 = 4by \) (which opens upwards)

### Step 2: Write the general equation of the tangent to the first parabola
The equation of the tangent to the parabola \( y^2 = 4ax \) can be expressed as:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent.

### Step 3: Substitute the tangent equation into the second parabola
Now, we substitute \( y = mx + \frac{a}{m} \) into the second parabola's equation \( x^2 = 4by \):
\[ x^2 = 4b\left(mx + \frac{a}{m}\right) \]
This simplifies to:
\[ x^2 = 4bmx + \frac{4ab}{m} \]

### Step 4: Rearrange into a standard quadratic form
Rearranging the equation gives us:
\[ x^2 - 4bmx - \frac{4ab}{m} = 0 \]

### Step 5: Apply the condition for tangency
For the line to be a common tangent, the quadratic equation must have exactly one solution. This occurs when the discriminant is zero:
\[ D = (4bm)^2 - 4 \cdot 1 \cdot \left(-\frac{4ab}{m}\right) = 0 \]
Calculating the discriminant:
\[ 16b^2m^2 + \frac{16ab}{m} = 0 \]

### Step 6: Solve the discriminant equation
Multiplying through by \( m \) (assuming \( m \neq 0 \)):
\[ 16b^2m^3 + 16ab = 0 \]
Factoring out \( 16 \):
\[ b^2m^3 + ab = 0 \]
This gives us:
\[ m^3 = -\frac{a}{b^2} \]

### Step 7: Find the value of \( m \)
Thus, we have:
\[ m = -\sqrt[3]{\frac{a}{b^2}} \]

### Step 8: Substitute \( m \) back into the tangent equation
Now we substitute \( m \) back into the tangent equation:
\[ y = -\sqrt[3]{\frac{a}{b^2}}x + \frac{a}{-\sqrt[3]{\frac{a}{b^2}}} \]

### Step 9: Simplify the tangent equation
This gives us the equation of the common tangents:
\[ y = -\sqrt[3]{\frac{a}{b^2}}x - \frac{ab^{2/3}}{a^{1/3}} \]

### Final Result
The equations of the common tangents to the parabolas \( y^2 = 4ax \) and \( x^2 = 4by \) are:
\[ y = mx + c \quad \text{where} \quad m = -\sqrt[3]{\frac{a}{b^2}} \quad \text{and} \quad c = -\frac{ab^{2/3}}{a^{1/3}} \]