Let the equation of two diameters of a circle x ^ 2 + y ^ 2 - 2x +...

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)\text{and }\left(2\right) we get,f = 0 or - 3$</p><p>Now given, Hyperbola$3{x}^{2} - {y}^{2} = 3$\text{or}{x}^{2} - \dfrac{{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)$⇒ 0 = m \pm \sqrt{{m}^{2} - 3}$

$⇒ m$ tends $\infty</p><p>Also given the$tangent passes through centre \left(1, 3\right)

So,$3 = m \pm \sqrt{{m}^{2} - 3} ⇒ {\left(3 - m\right)}$^{2} = {m}^{2} - 3$</p><p>$⇒ m = 2$</p></blockquote>$