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]}