How to write a system of equation that satisfies the conditions "a hyperbola and a circle that intersect in three points?"

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Answer 1 · 2016-11-12T00:15:57

${ (x^2-y^2 = 1), ((x-1)^2+y^2 = 2^2) :}$

The circle needs to touch the hyperbola at one point and cut it at the other two.

Choose a nice simple hyperbola:

$x^2-y^2 = 1$

This will intersect the $x$ axis at $(-1, 0)$ and $(1, 0)$

Then add a circle with centre $(1, 0)$ and radius $2$ :

$(x-1)^2+y^2 = 2^2$

These will intersect at $(-1, 0)$ , $(2, sqrt(3))$ and $(2, -sqrt(3))$

graph{(x^2-y^2-1)((x-1)^2+y^2-3.9)((x+1)^2+y^2-0.01)((x-2)^2+(y-sqrt(3))^2-0.01)((x-2)^2+(y+sqrt(3))^2-0.01) = 0 [-5, 5, -2.5, 2.5]}