"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 $2 x+p y=4 p$. Then the slope m $\in$ $(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to _______________."

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Accepted Answer

"$$ \begin{aligned} &2 p+f-1=0 \quad\dots(1)\\\\ &2-p f-4 p=0 \quad\dots(2)\\\\ &2=p(f+4) \\\\ &p=\frac{2}{f+4} \\\\ &2 p=1-f \\\\ &\frac{4}{f+4}=1-f \\\\ &f^2+3 f=0 \\\\ &f=0 \text { or }-3 \end{aligned} $$

Hyperbola $3 x^2-y^2=3, x^2-\frac{y^2}{3}=1$

$\mathrm{y}=\mathrm{mx} \pm \sqrt{\mathrm{m}^2-3}$

It passes $(1,0)$

$o=m \pm \sqrt{m^2-3}$

$m$ tends $\infty$

$$ \begin{aligned} &\text {It passes }(1,3) \\\\ &3=m \pm \sqrt{m^2-3} \\\\ &(3-m)^2=m^2-3 \\\\ &m=2 \end{aligned} $$"

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 $2 x+p y=4 p$. Then the slope m $\in$ $(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to _______________.

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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 $2 x+p y=4 p$. Then the slope m $\in$ $(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to _______________.

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