The locus of the mid-point of the chords of the hyperbola `x^(2)-y^(2)=4`, that touches the parabola `y^(2)=8x` is

Options

A. `x^(2)(x-2)=y^(3)`
B. `y^(2)(x-2)=x^(3)`
C. `x^(3)(x-2)=y^(2)`
D. `y^(3)(x-2)=x^(2)`

Answer

To find the locus of the mid-point of the chords of the hyperbola \(x^2 - y^2 = 4\) that touch the parabola \(y^2 = 8x\), we can follow these steps:

### Step 1: Define the midpoint of the chord
Let the midpoint of the chord be \(H, K\). The coordinates of the midpoint of the chord can be represented as \((H, K)\).

### Step 2: Write the equation of the chord
The equation of the chord of the hyperbola can be expressed as:
\[ x \cdot H - y \cdot K = H^2 - K^2 \]
This is derived from the general form of the chord of a conic section.

### Step 3: Rearrange the equation of the chord
Rearranging the equation gives:
\[ y = \frac{H}{K} x + \frac{H^2 - K^2}{K} \]
Let this equation be Equation (1).

### Step 4: Identify the tangent condition for the parabola
The line \(y = mx + c\) is tangent to the parabola \(y^2 = 8x\) if:
\[ c = \frac{4}{m} \]
In our case, \(m = \frac{H}{K}\) and \(c = \frac{H^2 - K^2}{K}\).

### Step 5: Substitute \(m\) and \(c\) into the tangent condition
Substituting \(m\) and \(c\) into the tangent condition gives:
\[ \frac{H^2 - K^2}{K} = \frac{4K}{H} \]

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying yields:
\[ (H^2 - K^2)H = 4K^2 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ H^3 - K^2H - 4K^2 = 0 \]

### Step 8: Factor out \(K^2\)
Rearranging further, we can express this as:
\[ K^2(H - 4) = H^3 \]

### Step 9: Find the locus
This implies that:
\[ y^2 (H - 4) = H^3 \]
This can be rewritten as:
\[ y^2 = \frac{H^3}{H - 4} \]

### Final Equation
Thus, the locus of the mid-point of the chords of the hyperbola that touch the parabola is given by:
\[ y^2 (x - 2) = x^3 \]

### Conclusion
The required locus is:
\[ y^2 (x - 2) = x^3 \]