If the angle between a pair of tangents drawn from a point p to. the circle $x^2+y^2-4x+2y+3=0\text{ is }\pi /2$ then

Let $P=\left(x_1,y_1\right).$ The equation to the pair of tangents from P to circle $x^2+y^2-4x+2y+3=0$ is

${\left[x{x}_1+y{y}_1-2\left(x+x_1\right)+1\left(y+y_1\right)+3\right]}^2=\left(x_1^2+y_1^2-4x_1+2y_1+3\right)\left(x^2+y^2-4x+2y+3\right)$

Angle between the tangents

$\Rightarrow 2\left(x_1^2+y_1^2-4x_1+2y_1+3\right)-{\left(x_1-2\right)}^2-{\left(y_1+1\right)}^2=0\Rightarrow x_1^2+y_1^2-4x_1+2y_1+1=0$

The locus of  P is $x^2+y^2-4x+2y+1=0$