Total number of common tangents of `y^(2) = 4ax " and " xy = c^(2)` is / are

Answer

To find the total number of common tangents between the parabola \( y^2 = 4ax \) and the hyperbola \( xy = c^2 \), we can follow these steps:

### Step 1: Write the equation of the tangent to the 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.

**Hint:** Remember that the slope \( m \) can vary, which will affect the position of the tangent line.

### Step 2: Write the equation of the tangent to the hyperbola
For the hyperbola \( xy = c^2 \), we can write the equation of the tangent as:
\[ y = mx + p \]
where \( p \) is a constant.

**Hint:** The tangent line to the hyperbola will also have the same slope \( m \) since we are looking for common tangents.

### Step 3: Set the two tangent equations equal
Since the tangents are common, we can equate the two equations:
\[ mx + \frac{a}{m} = mx + p \]
This simplifies to:
\[ \frac{a}{m} = p \]

**Hint:** This equality shows that the constant term of the tangent line for the parabola is equal to that of the hyperbola.

### Step 4: Substitute \( p \) in the hyperbola's tangent equation
Now substitute \( p \) in the hyperbola's tangent equation:
\[ xy = c^2 \implies x(mx + p) = c^2 \]
Substituting \( p = \frac{a}{m} \):
\[ x(mx + \frac{a}{m}) = c^2 \]
This expands to:
\[ mx^2 + a = c^2 \]

**Hint:** This equation is a quadratic in \( x \).

### Step 5: Set the discriminant to zero
For the line to be a tangent, the quadratic must have exactly one solution, which occurs when the discriminant is zero:
\[ b^2 - 4ac = 0 \]
Here, \( a = m \), \( b = 0 \), and \( c = a - c^2 \). Thus:
\[ 0^2 - 4m(a - c^2) = 0 \]
This simplifies to:
\[ 4m(a - c^2) = 0 \]

**Hint:** This condition means either \( m = 0 \) or \( a - c^2 = 0 \).

### Step 6: Analyze the conditions
1. If \( m = 0 \), the tangent is horizontal.
2. If \( a - c^2 = 0 \), then \( a = c^2 \).

**Hint:** Each condition gives us a specific scenario for the tangents.

### Step 7: Conclusion on the number of common tangents
From the analysis, we find that there is only one value of \( m \) that satisfies the conditions for common tangents. Thus, the total number of common tangents between the parabola and the hyperbola is:
\[ \text{Total number of common tangents} = 1 \]

**Final Answer:** The total number of common tangents is \( 1 \).