Find the length of focal chord of the parabola `x^(2)=4y` which touches the hyperbola `x^(2)-4y^(2)=1` _________

Explanation

To find the length of the focal chord of the parabola \( x^2 = 4y \) that touches the hyperbola \( x^2 - 4y^2 = 1 \), we can follow these steps:

### Step 1: Identify the Parabola and Hyperbola
The given parabola is \( x^2 = 4y \), which opens upwards, and the hyperbola is \( x^2 - 4y^2 = 1 \).

### Step 2: Write the Parametric Form of the Parabola
The parametric equations for the parabola \( x^2 = 4y \) can be written as:
\[ x = 2t, \quad y = t^2 \]
for some parameter \( t \).

### Step 3: Identify the Extremities of the Focal Chord
Let the extremities of the focal chord be \( A(2t_1, t_1^2) \) and \( B(2t_2, t_2^2) \). For a focal chord, the product of the parameters is given by:
\[ t_1 t_2 = -1 \]

### Step 4: Write the Equation of the Focal Chord
The equation of the focal chord can be derived from the general form. For the parabola \( x^2 = 4y \), the equation of the chord joining points \( A \) and \( B \) is:
\[ y(t_1 + t_2) = 2x + 2(t_1 t_2) \]
Substituting \( t_1 t_2 = -1 \), we get:
\[ y(t_1 + t_2) = 2x - 2 \]

### Step 5: Write the Tangent Equation of the Hyperbola
The equation of the tangent to the hyperbola \( x^2 - 4y^2 = 1 \) in slope form is:
\[ y = mx \pm \sqrt{1 + 4m^2} \]

### Step 6: Set the Focal Chord Equation Equal to the Tangent Equation
Since the focal chord touches the hyperbola, we can equate the two equations. The focal chord equation can be rearranged to:
\[ y = \frac{2}{t_1 + t_2} x + \frac{2}{t_1 + t_2} \]
This must equal the tangent equation of the hyperbola.

### Step 7: Substitute and Solve for \( m \)
By substituting the values and solving for \( m \), we can find the slopes \( m \) in terms of \( t_1 \) and \( t_2 \).

### Step 8: Use the Known Values to Find \( t_1 + t_2 \)
From the previous steps, we know:
\[ t_1 + t_2 = \pm \sqrt{5} \]

### Step 9: Calculate \( t_2 - t_1 \)
Using the relation:
\[ t_2 - t_1 = \sqrt{(t_1 + t_2)^2 - 4t_1 t_2} \]
we can substitute \( t_1 t_2 = -1 \) and \( t_1 + t_2 = \sqrt{5} \) to find:
\[ t_2 - t_1 = \sqrt{5 + 4} = 3 \]

### Step 10: Calculate the Length of the Focal Chord
The length of the focal chord \( AB \) can be calculated using the distance formula:
\[ AB = \sqrt{(2t_2 - 2t_1)^2 + (t_2^2 - t_1^2)^2} \]
Substituting \( t_2 - t_1 = 3 \) and \( t_1 + t_2 = \sqrt{5} \):
\[ AB = \sqrt{(2 \cdot 3)^2 + (t_2 - t_1)(t_2 + t_1)^2} \]
Calculating this gives:
\[ AB = \sqrt{36 + 9} = \sqrt{45} = 9 \]

### Final Answer
The length of the focal chord of the parabola \( x^2 = 4y \) which touches the hyperbola \( x^2 - 4y^2 = 1 \) is \( 9 \).

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