Question

If a circle intersects the hyperbola y = 1/x at four distinct points (xi , yi), i = 1, 2, 3, 4, then prove that x1x2 = y3y4.

Answer 1

Answer:

x1x2=y3y4

Step-by-step explanation:

We know the equation of a circle is (x-h)^2+(y-k)^2=r^2

replacing y with 1/x

(x-h)^2+((1/x)-k)^2=r^2

Rearranging all the terms

x^4-(2h)x^3+(h^2+k^2-r^2)x^2-(2k)x+1=0

We now know that x1,x2,x3,x4 are roots of the equation

so x1 X x2 X x3 X x4=1

we can also write this as

x1 X x2 X (1/y3) X (1/y4)=1

Thus getting the answer

x1 X x2 = y3 X y4