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 `x^2+y^2=b^2` (b) `x^2+y^2=a^2` `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

Answer

C

Explanation

The fourth vertex of the parallelogram lies on the circumcircle.
Therefore, the parallelogram is cyclic,
i.e., the parallelogram is a rectangle,
i.e., the tangents are perpendicular.
Therefore, the locus of P is the director circle.