From points on the circle `x^2+y^2=a^2` tangents are drawn to the hyperbola `x^2-y^2=a^2`. Then, the locus of mid-points of the chord of contact of tangents is:

Options

A. `(x^2-y^2)^2=a^2(x^2+y^2)`
B. `(x^2-y^2)=a^2(x^2+y^2)`
C. `(x^2-y^2)=a^2(x^2+y^2)^2`
D. None of these

Answer

To find the locus of midpoints of the chord of contact of tangents drawn from points on the circle \(x^2 + y^2 = a^2\) to the hyperbola \(x^2 - y^2 = a^2\), we can follow these steps:

### Step 1: Identify a point on the circle
Let the point \(P\) on the circle be represented in parametric form as:
\[ P(a \cos \theta, a \sin \theta) \]

### Step 2: Write the equation of the chord of contact
The equation of the chord of contact from point \(P\) to the hyperbola can be expressed as:
\[ x \cos \theta - y \sin \theta = a \]

### Step 3: Define the midpoint of the chord
Let the midpoint of the chord be \(M(h, k)\). The equation of the chord can also be expressed in terms of the midpoint \(M\):
\[ hx - ky = h^2 - k^2 \]

### Step 4: Relate the two equations
From the equations of the chord of contact and the midpoint, we can set them equal:
\[ x \cos \theta - y \sin \theta = hx - ky \]
This implies that the coefficients of \(x\) and \(y\) must be equal. Therefore, we have:
\[ \cos \theta = h \quad \text{and} \quad -\sin \theta = -k \]

### Step 5: Use the identity for sine and cosine
From the above, we can express:
\[ \cos^2 \theta + \sin^2 \theta = 1 \implies h^2 + k^2 = 1 \]

### Step 6: Substitute back to find the locus
Now we need to relate this back to the original equations. Since \(P\) lies on the circle, we can relate \(h\) and \(k\) to the original equations of the circle and hyperbola.

From the hyperbola, we know:
\[ h^2 - k^2 = a^2 \]
Substituting \(h^2 + k^2 = a^2\) into this gives:
\[ h^2 - k^2 = a^2 \implies h^2 = \frac{a^2 + k^2}{2} \quad \text{and} \quad k^2 = \frac{a^2 - h^2}{2} \]

### Step 7: Form the final equation
Combining these, we can derive the locus of the midpoints \(M(h, k)\) as:
\[ h^2 - k^2 = a^2(h^2 + k^2) \]
This leads us to the final locus equation:
\[ h^2 - k^2 = a^2 \]

### Conclusion
Thus, the required locus of the midpoints of the chord of contact of tangents drawn from points on the circle to the hyperbola is given by:
\[ h^2 - k^2 = a^2 \]