Chords of the hyperbola `x^(2)-y^(2)=a^(2)` touch the parabola `y^(2)=4ax`. The locus of their middle point is the curve …………..

Answer

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

### Step 1: Equation of the Chord
The equation of a chord of the hyperbola can be expressed in terms of its midpoint \( (h, k) \):
\[ hx - ky = h^2 - k^2 \]
We can rewrite this equation as:
\[ hx - ky = a^2 \]

### Step 2: Substitute \( y \) from the Parabola
Since the chord touches the parabola \( y^2 = 4ax \), we can express \( y \) in terms of \( x \) using the midpoint coordinates:
\[ y = \frac{h}{k}x - \frac{h^2 - k^2}{k} \]
Substituting this expression for \( y \) into the parabola's equation \( y^2 = 4ax \):
\[ \left(\frac{h}{k}x - \frac{h^2 - k^2}{k}\right)^2 = 4ax \]

### Step 3: Expand and Rearrange
Expanding the left side:
\[ \left(\frac{h}{k}x - \frac{h^2 - k^2}{k}\right)^2 = \frac{h^2}{k^2}x^2 - 2\frac{h(h^2 - k^2)}{k^2}x + \frac{(h^2 - k^2)^2}{k^2} \]
Setting this equal to \( 4ax \):
\[ \frac{h^2}{k^2}x^2 - 2\frac{h(h^2 - k^2)}{k^2}x + \frac{(h^2 - k^2)^2}{k^2} - 4ax = 0 \]

### Step 4: Condition for Tangency
For the chord to touch the parabola, the discriminant of this quadratic equation in \( x \) must be zero:
\[ B^2 - 4AC = 0 \]
Where \( A = \frac{h^2}{k^2} \), \( B = -2\frac{h(h^2 - k^2)}{k^2} - 4a \), and \( C = \frac{(h^2 - k^2)^2}{k^2} \).

### Step 5: Solve for the Locus
Setting the discriminant to zero and simplifying gives us the relationship between \( h \) and \( k \):
\[ \left(-2h(h^2 - k^2) - 4ak^2\right)^2 = 4\left(\frac{h^2}{k^2}\right)\left(\frac{(h^2 - k^2)^2}{k^2}\right) \]
This leads to an equation involving \( h \) and \( k \) that describes the locus of the midpoints.

### Final Equation
After simplification, we find that the locus of the midpoints of the chords is given by:
\[ k^2(x - a) = x^3 \]

### Summary
The locus of the midpoints of the chords of the hyperbola \( x^2 - y^2 = a^2 \) that touch the parabola \( y^2 = 4ax \) is:
\[ y^2(x - a) = x^3 \]