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Let's say I have a circumference with the equation $x^2 + y^2-10=0$. This circumference has a point A $(1;3)$ which which passes thorough an equilateral hyperbola $xy=3$.

I would like to find all the intersections. I have already found one of them $A(1;3)$. In order to find the other one I put the equation of the circumference and the one of the hyperbola together in a system. How do I find all the intersections?

Cesare
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Right, put them in a system: $xy=3$ and $x^2+y^2=10\Leftrightarrow (x+y)^2-2xy=10\Leftrightarrow (x+y)^2=10+2.3=16\Rightarrow x+y=4$ or $x+y=-4$. Now you have to solve the two resulting systems:

$xy=3,\,\,x+y=4$ and $xy=3,\,\,x+y=-4$

Svetoslav
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$$x^2+y^2-10=0\\xy=3.$$

Then multiplying by $x^2$,

$$x^4+x^2y^2-10x^2=x^4-10x^2+9=(x-1)(x+1)(x-3)(x+3).$$

The solutions are $(-3,-1),(-1,-3),(1,3),(3,1).$