Let the equation of two diameters of a circle x 2 + y 2 - 2 x + 2 f y + 1 = 0 be 2 p x - y = 1 and 2 x + p y…

Given, 

Equation of circle $x^2+y^2-2x+2fy+1=0$ and given diametric lines will pass through $\left(1,-f\right)$ which is the centre of the circle, so $2p+f-1=0 ...\left(1\right)$ and $2-pf-4p=0 ...\left(2\right)$

Now solving equation $\left(1\right) \& \left(2\right)$ we get,  $f=0$ or $-3$

Now given, Hyperbola $3x^2-y^2=3 \text{or} x^2-\frac{y^2}{3}=1$

We know that tangent to hyperbola is given by $y=mx\pm \sqrt{m^2-3}$

It passes $\left(1,0\right)$ $\Rightarrow 0=m\pm \sqrt{m^2-3}$

$\Rightarrow m$ tends $\infty$

Also given the tangent passes through centre $\left(1,3\right)$

So, $3=m\pm \sqrt{m^2-3}$ $\Rightarrow {\left(3-m\right)}^2=m^2-3$

$\Rightarrow m=2$