Let `y^(2)=4ax` be a parabola and `x^(2)-y^(2)=a^(2)` be a hyperbola. Then number of common tangents is

Options

A. 2 for `alt0`
B. 1 for `alt0`
C. 2 for `agt0`
D. 1 for `agt0`

Answer

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

### Step 1: Write the equations of the curves
The equations given are:
1. Parabola: \( y^2 = 4ax \)
2. Hyperbola: \( x^2 - y^2 = a^2 \)

### Step 2: Write the equation of the tangent to the parabola
The equation of the tangent to the parabola \( y^2 = 4ax \) at a slope \( m \) is given by:
\[ y = mx + \frac{a}{m} \]

### Step 3: Write the equation of the tangent to the hyperbola
The equation of the tangent to the hyperbola \( x^2 - y^2 = a^2 \) at a slope \( m \) is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - a^2} \]
This can be simplified to:
\[ y = mx \pm a\sqrt{m^2 - 1} \]

### Step 4: Set the tangents equal to find common tangents
For the tangents to be common, the two equations must be equal:
\[ mx + \frac{a}{m} = mx \pm a\sqrt{m^2 - 1} \]
By eliminating \( mx \) from both sides, we have:
\[ \frac{a}{m} = \pm a\sqrt{m^2 - 1} \]

### Step 5: Solve for \( m \)
Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ \frac{1}{m} = \pm \sqrt{m^2 - 1} \]
Squaring both sides gives:
\[ \frac{1}{m^2} = m^2 - 1 \]
Multiplying through by \( m^2 \) leads to:
\[ 1 = m^4 - m^2 \]
Rearranging gives us:
\[ m^4 - m^2 - 1 = 0 \]

### Step 6: Let \( t = m^2 \)
Letting \( t = m^2 \), we rewrite the equation as:
\[ t^2 - t - 1 = 0 \]

### Step 7: Use the quadratic formula
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2} \]
This gives us two positive values for \( t \), which correspond to two values of \( m \).

### Step 8: Determine the number of common tangents
Since we have two distinct positive values of \( m^2 \), we can conclude that there are two distinct slopes for the tangents. Each slope corresponds to two tangents (one for each sign in the hyperbola's tangent equation).

Thus, the total number of common tangents is:
\[ \boxed{2} \]