Question

If a circle intersects the hyperbola $y=\frac{1}{x}$ at four distinct points $(x_i,y_i),i=1,2,3,4,$ then find the value of $x_1{x}_2$ - $y_3{y}_4$

Answer 1

We have to solve the system $(x-h{)}^2+(y-k{)}^2=r^2$ and
$y=\frac{1}{x}$
In the equation of the circle above, h, k, and r are all constants. The problem also assures us that the circle and the hyperbola will intersect in 4 distinct points.
Substitute the bottom equation into the top equation:
$(x-h{)}^2+(\frac{1}{x}-k{)}^2=r^2$
Clear fractions:
$x^4-2h{x}^3+h^2{x}^2+1-2kx+k^2{x}^2-r^2{x}^2=0$
Combine like terms:
$x^4-2h{x}^3+(h^2+k^2-r^2)x^2-2kx+1=0$ <---- (A)
From algebra, we know that for the polynomial
$a_0{x}^n+a_1{x}^n-1+.....+a_k{x}^n-k+......+a_n-2x^n-2+a_n-1x^n-1+a_n=0,$
$(-1{)}^k{a}_k$ is equal to the summation of product of the roots of the polynomial taken k at a time. We need this result only for the constant term of (A).
We get $(-1{)}^4{x}_1.x_2.x_3.x_4=1$
This is the same as $x_1.x_2.\left(\frac{1}{y_3}\right).\left(\frac{1}{y_4}\right)=1$, or finally,
$x_1.x_2=y_3.y_4$

Thus $x_1.x_2-y_3.y_4=0$

Hence answer is 0.