Find the length of that focal chord of the parabola `y^(2) = 4ax`, which touches the rectangular hyperbola `2xy = a^(2)` .

Answer

To find the length of the focal chord of the parabola \( y^2 = 4ax \) that touches the rectangular hyperbola \( 2xy = a^2 \), we can follow these steps:

### Step 1: Understand the Parabola and Hyperbola
The parabola \( y^2 = 4ax \) opens to the right with its focus at \( (a, 0) \). The rectangular hyperbola \( 2xy = a^2 \) can be rewritten as \( xy = \frac{a^2}{2} \).

### Step 2: Parametric Representation of the Parabola
The points on the parabola can be represented parametrically as:
\[ P_1 = (at_1^2, 2at_1) \quad \text{and} \quad P_2 = (at_2^2, 2at_2) \]
where \( t_1 \) and \( t_2 \) are parameters corresponding to the points on the parabola.

### Step 3: Condition for the Focal Chord
For a focal chord, the product of the parameters \( t_1 \) and \( t_2 \) must satisfy:
\[ t_1 t_2 = -1 \]

### Step 4: Find the Tangent to the Hyperbola
The equation of the hyperbola \( 2xy = a^2 \) can be differentiated to find the slope of the tangent:
\[ \frac{dy}{dx} = -\frac{a^2}{2y} \]
At the point \( (x_0, y_0) \) on the hyperbola, the tangent line can be expressed as:
\[ y - y_0 = m(x - x_0) \]
where \( m \) is the slope.

### Step 5: Substitute Points into the Hyperbola
Substituting the coordinates of the points \( P_1 \) and \( P_2 \) into the hyperbola equation gives:
\[ 2(at_1^2)(2at_1) = a^2 \quad \text{and} \quad 2(at_2^2)(2at_2) = a^2 \]
This leads to the equations:
\[ 4a^2 t_1^3 = a^2 \quad \Rightarrow \quad t_1^3 = \frac{1}{4} \quad \Rightarrow \quad t_1 = \frac{1}{\sqrt[3]{4}} \]
and similarly for \( t_2 \).

### Step 6: Calculate Length of the Focal Chord
The length of the focal chord can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(at_1^2 - at_2^2)^2 + (2at_1 - 2at_2)^2} \]
Substituting \( t_1 \) and \( t_2 \) into this formula and simplifying will yield the length of the focal chord.

### Step 7: Final Calculation
After substituting the values and simplifying, we find:
\[ \text{Length} = 5a \]

### Conclusion
Thus, the length of the focal chord of the parabola \( y^2 = 4ax \) that touches the rectangular hyperbola \( 2xy = a^2 \) is \( 5a \).

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