If tangents `P Qa n dP R` are drawn from a variable point `P` to thehyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1,(a > b),` so that the fourth vertex `S` of parallelogram `P Q S R` lies on the circumcircle of triangle `P Q R` , then the locus of `P` is (a) `x^2+y^2=b^2` (b) `x^2+y^2=a^2` (c) `x^2+y^2=a^2-b^2` (d) none of these

Options

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

Explanation

To find the locus of the point \( P \) from which tangents \( PQ \) and \( PR \) are drawn to the hyperbola given by

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > b) \]

and where the fourth vertex \( S \) of the parallelogram \( PQSR \) lies on the circumcircle of triangle \( PQR \), we can follow these steps:

### Step 1: Understanding the Geometry
The tangents \( PQ \) and \( PR \) are drawn from point \( P \) to the hyperbola. The point \( S \) is the fourth vertex of the parallelogram formed with points \( P, Q, \) and \( R \). Since \( S \) lies on the circumcircle of triangle \( PQR \), we can conclude that the quadrilateral \( PQRS \) is cyclic.

### Step 2: Properties of Cyclic Quadrilaterals
For a cyclic quadrilateral, the opposite angles sum up to \( 180^\circ \). This implies that the tangents \( PQ \) and \( PR \) must be perpendicular to each other.

### Step 3: Condition for Perpendicular Tangents
The condition for the tangents from a point \( P(x_1, y_1) \) to the hyperbola to be perpendicular is given by:

\[ \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1 \]

However, since we are looking for the locus of point \( P \), we need to express this condition in terms of \( x \) and \( y \).

### Step 4: Using the Director Circle
The locus of the point \( P \) where the tangents are perpendicular is actually the equation of the director circle of the hyperbola. The equation of the director circle for the hyperbola is given by:

\[ x^2 + y^2 = a^2 - b^2 \]

### Step 5: Conclusion
Thus, the locus of point \( P \) is:

\[ x^2 + y^2 = a^2 - b^2 \]

This corresponds to option (c).

### Final Answer
The locus of \( P \) is:

\[ \boxed{x^2 + y^2 = a^2 - b^2} \]