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
Let the midpoint of the chord be \(P(h, k)\), where \(h\) and \(k\) are the coordinates of the midpoint.

### Step 2: Write the Equation of the Chord
The equation of the chord of the hyperbola \(x^2 - y^2 = 4\) with midpoint \(P(h, k)\) can be expressed as:
\[ x h - y k = h^2 - k^2 \]
This represents the chord of the hyperbola that passes through the point \(P(h, k)\).

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ k y = h x - (h^2 - k^2) \]
This can be rewritten as:
\[ y = \frac{h}{k} x - \frac{h^2 - k^2}{k} \]

### Step 4: Condition for Tangency
Since this chord touches the parabola \(y^2 = 8x\), we need to apply the condition for tangency. For the line \(y = mx + c\) to be tangent to the parabola \(y^2 = 8x\), the discriminant of the resulting quadratic equation must be zero.

Substituting \(y = \frac{h}{k} x - \frac{h^2 - k^2}{k}\) into \(y^2 = 8x\):
\[ \left(\frac{h}{k} x - \frac{h^2 - k^2}{k}\right)^2 = 8x \]

### Step 5: Expanding and Rearranging
Expanding the left-hand side:
\[ \frac{h^2}{k^2} x^2 - 2\frac{h(h^2 - k^2)}{k^2} x + \frac{(h^2 - k^2)^2}{k^2} = 8x \]
Rearranging gives:
\[ \frac{h^2}{k^2} x^2 - \left(2\frac{h(h^2 - k^2)}{k^2} + 8\right)x + \frac{(h^2 - k^2)^2}{k^2} = 0 \]

### Step 6: Setting the Discriminant to Zero
For tangency, the discriminant must be zero:
\[ \left(2\frac{h(h^2 - k^2)}{k^2} + 8\right)^2 - 4\left(\frac{h^2}{k^2}\right)\left(\frac{(h^2 - k^2)^2}{k^2}\right) = 0 \]

### Step 7: Simplifying the Condition
This leads to a relationship between \(h\) and \(k\). After simplifying, we can express \(k^2\) in terms of \(h\):
\[ k^2(h - 2) = h^3 \]

### Step 8: Final Locus Equation
This gives us the locus of the midpoints as:
\[ k^2 = \frac{h^3}{h - 2} \]
Substituting \(k = y\) and \(h = x\), we get:
\[ y^2 = \frac{x^3}{x - 2} \]

### Final Result
Thus, the locus of the midpoints of the chords of the hyperbola that touch the parabola is:
\[ y^2 = \frac{x^3}{x - 2} \]