Explanation

(a)
We know that the locus of the point of intersection of perpendicular tangents to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is a circle $x^{2} + y^{2} = a^{2} - b^{2}.$
Thus, locus of the point of intersection of perpendicular tangents to the hyperbola $\frac{x^{2}}{3} - \frac{y^{2}}{1} = 1$ is a circle
$x^{2} + y^{2} = 3 - 1$
${\Rightarrow \ x}^{2} + y^{2} = 2$