If a circle intersects the hyperbola $y=1/x$ at four distinct points $(x_i,y_i), i=1,2,3,4,$ then prove that $x_1x_2=y_3y_4$.

Question

If a circle intersects the hyperbola $y=1/x$ at four distinct points $(x_i,y_i), i=1,2,3,4,$ then prove that $x_1x_2=y_3y_4$.

I have really no idea on how to approach this question.

One clumsy way might be to consider an arbitrary circle $(x-a)^2+(y-b)^2=r^2$, such that it intersects the hyperbola $y=1/x$ at four distinct points $(x_i,y_i), i=1,2,3,4,$ and then manipulate stuffs to finally get the equality $x_1x_2=y_3y_4$, though I am not sure.

Related : https://math.stackexchange.com/questions/3577217/concyclic-points-on-hyperbola — lab bhattacharjee, Mar 12 '20 at 07:46

score 2 · Accepted Answer · answered Mar 12 '20 at 07:31

take the point on hyperbola as (t,$\frac{1}{t})$. Put it in the equation of the circle, you will get a fourth degree equation where $t_i$ corresponds to $x_i$ and $\frac{1}{t_i}$ corresponds to $y_i$. Use theory of equations and you should get the result. It will be little easier if you consider the circle as $x^2+y^2=a^2$ but you may go ahead with general case also.

If a circle intersects the hyperbola $y=1/x$ at four distinct points $(x_i,y_i), i=1,2,3,4,$ then prove that $x_1x_2=y_3y_4$.

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